Systems and methods for an enterprise pricing solution

ABSTRACT

Embodiments of a system and method for an enterprise product line design and pricing solution are disclosed.

CROSS REFERENCE TO RELATED APPLICATIONS

This is a U.S. non-provisional patent application that claims benefit toU.S. provisional patent application Ser. No. 62/713,784 filed on Aug. 2,2018, which is incorporated by reference in its entirety.

FIELD

The present disclosure generally relates to an enterprise pricingsolution, and in particular to systems and methods for acomputer-implemented enterprise pricing solution to address thetechnical challenge of designing and computing pricing for complexproduct lines.

BACKGROUND

Firms frequently introduce new products and retire old products to renewcustomers' interests in purchasing and consuming new products. Forexample, restaurants regularly add new items to its menu to spur newinterests. Hotels introduce new room choices (e.g., free breakfastcombo, executive package, and so on) from time to time to attract morecustomers. Manufacturers of home appliances periodically launch newmodels with improved features and efficiency driven by new trends incustomer preference and life style. This phenomenon is even morepronounced in the high-tech industry due to a faster industryclockspeed. At Intel, for example, new microprocessor products arereleased on a quarterly basis. Such a product cadence leads toconstantly evolving product lines and, at each change epoch, a firm hasto determine what products to add to an existing product line and atwhat price points. That is, given the attributes and prices of theexisting products, the firm optimizes the attributes and/or prices ofthe new products to maximize the total profit from the product line.Such business decisions are plagued by multiple complications. Higherattribute values and lower prices increase product appeal and attractmore customers, but at a cost; attributes are costly and lower pricesdecrease unit revenue, both of which contribute to lower margins. Inaddition, the attractiveness of new products affects the market share ofexisting products and may cannibalize the profit of existing products.Thus, the firm needs to strike a balance between these competing forces.

This decision problem arises in a variety of industries but the natureof the problem is similar across industries. Consider a resort hotelthat is adding new room choices to its existing offerings. Managementwould like to offer value packages that include basic room service plusresort credit to be used for ancillary services such as restaurants,gift shop, spa and entertainment. Table 1 provides information onexisting products and a plausible set of new product offerings. Thehotel's decision problem is to set the proper resort credit level and/orprice for each new offer.

TABLE 1 Room Offerings at a Resort Hotel. Resort Resort Existing RoomCredit Price New Room Credit Price 1 One 0 259 4 One 50 289 King King 2Two 0 299 5 Two 80 359 Queen Queen 3 King 0 359 6 King 100 429 SuiteSuite

In another example, a smart phone manufacturer introduces a new model(M2) of phones to its product line, which will be sold concurrently withan existing phone model (M1). The manufacturer offers severalstorage-size variations of each model and needs to determine the storagesize and/or price of each variation of the new model (Table 2).

TABLE 2 Existing and New Phone Models for a Smartphone ManufacturerExisting Model Storage Price New Model Storage Price 1 M1 16 149 4 M2 64299 2 M1 32 199 5 M2 128 349 3 M1 64 269 6 M2 256 499

In both examples, the attribute to be optimized is a dimension thatvertically differentiates the products and the firm optimizes the newproducts in the presence of existing products.

It is with these observations in mind, among others, that variousaspects of the present disclosure were conceived and developed.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A and 2B are graphical representations showing optimal attributevalues versus b and β;

FIGS. 2A and 2B are other graphical representations showing optimalattribute values versus b and β;

FIG. 3 is an example schematic diagram of a computing system that mayimplement various methodologies of systems and methods for an enterprisepricing solution; and

FIG. 4 is a simplified block diagram of a computing system and/ornetwork related to the computing device of FIG. 3 for implementingaspects of the functionality described herein.

Corresponding reference characters indicate corresponding elements amongthe view of the drawings. The headings used in the figures do not limitthe scope of the claims.

DETAILED DESCRIPTION

Various embodiments of a computer-implemented enterprise pricingsolution that outputs a pricing value defining a solution for companiesfacing the challenge of designing complex product lines and computingoptimized product line prices and attributes in an evolving product lineare disclosed.

Introduction

In the present disclosure, the attribute and price decisions forproducts with a vertically-differentiated attribute are disclosed. Theattribute in some specific cases may be the quality of a product.Specifically, three variations of the problem are considered that arerelevant in practice. In variation (i), the attribute values of allproducts are given and the firm optimizes the prices of the newproducts. This often arises when the attributes are pre-determinedduring the design and engineering stage and prices are decided close toproduct launch. In variation (ii), the prices of the new products areexogenous and the firm optimizes the product attributes. This isrelevant when the firm follows pre-determined price points either due toconvention or customer expectation. For example, it is a common practiceat Intel to offer certain products at fixed price points: between March2008 and August 2009, Intel repeatedly introduced its top product in theCore 2 family at the $530 price point. In variation (iii), the firmoptimizes both price and attribute of the new products. The jointdecision on prices and attributes typically occurs during strategicplanning when a firm plans the next generation of product offerings. Inpractice, variations (i)-(iii) may be adopted by the same firm fordifferent decision scopes and contexts. For instance, a firm may solve avariation (iii) problem during strategic planning but may re-optimizeprices later by solving a variation (i) problem as a tacticaladjustment. These decisions are considered when a firm adds to anexisting product line but we note that the “clean-slate” version of theproblem, i.e., when the firm can decide prices and/or attributes of allproducts to be offered, is a special case of the model in the presentdisclosure. This includes the case where the firm is starting a newproduct line, as well as the case in which the firm decides tore-optimize the prices and/or attributes of the existing products alongwith the new products.

Demand is modeled using a widely-adopted choice model for customersfacing multiple product options—the multinomial logit (MNL) model. A newfeature we include in the MNL model is the interaction between productattribute and product price, which allows the marginal utility of theattribute value to depend on the price level of the product, andlikewise, marginal utility of price to depend on the attribute value ofthe product. This captures the commonly observed phenomenon thatcustomers are less sensitive to price changes in high-quality productthan low-quality product. It will be shown that this interactionrationalizes the matching of high markup with high quality whichsupplements the equal-markup pricing result in the literature. Moresignificantly, it has important implications for decisions involvingproduct attribute. It is further shown that the sequence of optimalattribute values among the newly offered products match the sequence ofprices, controlling for other parameters; this holds in both attributeoptimization and joint optimization. Analysis indicates that theinteraction term plays a central role in justifying differentiatedoffering of new products. With the interaction effect of attribute andprice, the optimal attribute and markup vary across products even underidentical price sensitivity and cost function, which is acommonly-adopted strategy that can now be quantified and optimized withthe model developed in this disclosure.

The ease of use and interpretation of the model makes the interactionterm a simple but powerful tool for incorporating the moderating effectof price and quality on how they each affect customer utility or demand.This addition is built upon and the interaction term is adopted in theMNL model in a normative decision setting. This is the first applicationof this modification to a firm's product line decision under MNL demand.This interaction rationalizes the matching of high markup with highquality which supplements the equal-markup pricing result in theliterature. The equal-markup result derives from the orthogonality ofprice and quality in their effect on customer utility and is not alwaysconsistent with observations in practice. The practice of charginghigher markups for high quality is ubiquitous in today's market as well.For example, Apple currently sells two models of iPhone 7 at $549 and$649 for 32 GB and 128 GB capacity respectively, while the two models ofiPhone 8 with 64 GB and 256 GB capacity are selling for $699 and $849respectively (Apple Corporation Website, 2018b). Intel has a longhistory of selling its top-bin products (those with higher speedperformance) at a much higher margin than the lower-bin products (IntelCorporation Website, 2018). Therefore, the result presented in thepresent disclosure is a more realistic characterization of the pricingdecision. More significantly, it has important implications fordecisions involving product quality. It is show that the sequence ofoptimal quality values among the newly offered products matches thesequence of prices, controlling for other parameters. The analysisdescribed herein indicates that the interaction term plays a centralrole in justifying differentiated offering of new products. With theinteraction effect of price and quality, the optimal quality and markupvary across products even under identical price sensitivity and costfunction.

The theoretical contributions are fourfold: First, it is the firstsolution to solve joint pricing and attribute decisions under the MNLmodel allowing continuous attribute values. Existing literature thataddresses joint pricing and attribute decisions examines a givenassortment of products, i.e., the attribute values are a finite set ofpre-selected discrete values and thus the insights are limited toassortment selection. The present disclosure, in contrast, yieldsstrategic insights on how firm should design its product line andoptimally set product attribute values on a continuum in conjunctionwith prices, providing decision support with a new dimension. Concavityof the profit functions is established under separate price andattribute optimizations and with considerations of existing products inthe product line and we identify a sufficient condition for uniqueoptimal solution for the joint price-attribute optimization. Second, theoptimal prices and attributes are characterized and efficient algorithmsdeveloped for each problem variation. Third, the present disclosure isthe first to include price-quality interaction in the optimization whichhelps reconcile the divergence of existing literature's equal-markupprice prediction from empirical practices and uncovers new insights onproduct attribute decisions as the product line evolves. Lastly, thedisclosed model extends to a multi-attribute setting for both attributeoptimization and joint price-attribute optimization.

Model

A customer makes a selection of one of n product choices and ano-purchase alternative. The product purchase probabilities are given bythe MNL model. Let the utility of product i, i=1, 2, . . . , n be

u _(i) =x _(i) −b _(i) p _(i)+β_(i) x _(i) p _(i) +a _(i) +ϵ _(i)

where x_(i) is the attribute value, p_(i) is the price of product i,a_(i) represents an observable utility term that is independent of x_(i)and p_(i), and ϵ_(i) is a random noise term which is a Gumbel randomvariable that represents unobserved utility. a_(i) refers to utilityfrom attributes that are exogenously determined and orthogonal to theattribute x_(i). For example, for hotel rooms, a_(i) may capture utilityassociated with the type of room such as King, Queen and Suite, whereasx_(i) reflects utility of ancillary services such as packages thatinclude resort credit, event activities and meals. For smart phoneproducts, a_(i) may be associated with the model type (e.g., iPhone 7,iPhone 7plus), and x_(i) may be associated with the storage size (e.g.,64 GB, 256 GB), similar to the example in Table 2. That is, the qualitymeasure x is a linear utility scale transformed from the nominal scaleof a certain attributes such as storage size or resort credit. Withoutloss of generality, it can be assumed that x_(i) ∈[0, x_(i) ⁺] wherex_(i)=0 and x_(i)=x_(i) ⁺ align with the lowest and highest possiblequality level respectively. For example, the quality scale for smartphones could be the logarithm transformation of the nominal storagesize. Then, adjusting for a minimum required size of 16 GB (i.e., alignx_(s)=0 with 16 GB), nominal values of 64 GB, 128 GB and 256 GBcorrespond to x values of 0.6, 0.9, and 1.2 respectively, while themaximum x_(i) ⁺ may correspond to the scaled value of some practicalupper limit of storage size. Parameters b_(i)>0 and β_(i)≥0 are thecoefficients for price sensitivity where b_(i) is quality-independentsensitivity and β_(i) is the coefficient for the interaction term andcaptures the heterogeneity in customer sensitivity towards the productprice at different quality levels. It is assumed that b_(i)−β_(i)x_(i)is always positive, i.e., customers always experience a disutilitytoward higher prices.

Interaction terms in regression models are used to capture how themarginal effect of one explanatory variable on the dependent variable ismodified by another explanatory variable and are prevalent in statisticsand econometric applications. An interaction term is typically modeledas the product of two variables in the regression equation

Y=β ₀+β₁ X ₁+β₂ X ₂+β₁₂ X ₁ X ₂+ϵ.

The same form of interaction terms is also commonly adopted in logit andprobit models. The interaction term in the utility function of the logitmodel allows the marginal utility of attribute x to depend on price pand equivalently, the marginal disutility of price p to depend on x.Specifically, rewrite the utility function in two alternative forms.

u _(i) =x _(i)−(b _(i)−β_(i) x _(i))p _(i) +a _(i)+ϵ_(i) and  (1)

u _(i)=(1+β_(i) p _(i))x _(i) −b _(i) p _(i) +a _(i)+ϵ_(i).  (2)

The marginal disutility of p_(i) is given by (b_(i)−β_(i)x_(i)) whichdecreases with attribute; the marginal utility of x_(i) is (1+β_(i) p_(i)) which increases with price. As discussed previously, this ineffect models the empirical observation that customers are lesssensitive to price change at high quality, or equivalently, customersare more sensitive to quality change at high price (i.e.,

${{- \frac{\partial}{\partial x_{i}}}\left( \frac{\partial\left( {- u_{i}} \right)}{\partial p_{i}} \right)} = {{\frac{\partial}{\partial p}\left( \frac{\partial u_{i}}{\partial x_{i}} \right)} \geq {0\mspace{14mu} {where}\mspace{14mu} \frac{\partial\left( {- {ui}} \right)}{\partial p_{i}}}}$

is the marginal disutility of price).

Let J be the set of existing products and l be the set of new products.Assume the no-purchase option has a utility of zero. For product j∈J,its price, attribute, and cost values are fixed at p _(j), x _(j), and c_(j) respectively. Let x=(x_(i))_(i∈l) be the vector of attribute valuesof the new products. For the ease of notation, we also define ū_(j)=x_(j)−b_(j) p _(j)+β_(j) x _(j) p _(j) + _(j), m _(j)=p _(j)−c _(j) and

${\overset{\_}{\pi}}_{J} = {\frac{\sum\limits_{j \in J}{{\overset{\_}{m}}_{j}e^{{\overset{\_}{u}}_{j}}}}{1 + {\sum\limits_{j \in J}{e^{u}j}}}.}$

Note that π _(J) is the firm's expected profit prior to the addition ofthe new products.

The purchase probability of a new product l∈l is

$\begin{matrix}{{q_{i} = \frac{e^{x_{i} - {b_{i}p_{i}} + {\beta_{i}x_{i}p_{i}} + \alpha_{i}}}{1 + {\sum\limits_{j \in J}e^{\overset{\_}{u}j}} + {\sum\limits_{i^{\prime} \in I}e^{x_{i^{\prime}} - {b_{i^{\prime}}p_{i^{\prime}}} + {\beta_{i^{\prime}}x_{i^{\prime}}p_{i^{\prime}}} + a_{i^{\prime}}}}}},} & (3)\end{matrix}$

the purchase probability of an existing product j∈J is

$\begin{matrix}{{q_{j} = {\frac{e^{{\overset{\_}{u}}_{j^{\prime}}}}{1 + {\sum\limits_{j^{\prime} \in J}e^{{\overset{\_}{u}}_{j^{\prime}}}} + {\sum\limits_{i \in I}e^{x_{i^{\prime}} - {b_{i}p_{i}} + {\beta_{i}x_{i}p_{i}} + a_{i}}}}\mspace{14mu} {and}}}{q_{0} = \frac{1}{1 + {\sum\limits_{j \in J}e^{{\overset{\_}{u}}_{j}}} + {\sum\limits_{i \in I}e^{x_{i} - {b_{i}p_{i}} + {\beta_{i}x_{i}p_{i}} + a_{i}}}}}} & (4)\end{matrix}$

is the no-purchase probability. Therefore,

q _(i) =q ₀ e ^(x) ^(i) ^(−b) ^(i) ^(p) ^(i) ^(+β) ^(i) ^(x) ^(i) ^(p)^(i) ^(+a) ^(i) and  (5)

q_(j)=q₀e^(ū) ^(j) .  (6)

Price Optimization

Price optimization arises when the firm sets prices of the new productsto maximize the total profit from the product line. For instance, inTable 1, the resort hotel decides the prices of the new room offersbased on the information of existing room offers and the planned servicepackages for each room type in the new offers; similarly in Table 2, thesmartphone manufacturer decides the prices of the new phones given allother information. The present disclosure provides solutions to thisproblem through two features: (1) consideration of existing products inthe product line, and (2) consideration of quality-price interaction. Inthe presence of existing products, the pricing decision of the newproducts affects not only the relative market share of the new products,but also those of the existing products. Hence, it does not merely implyan enlarged no-purchase utility with the additional constant termΣ_(j∈J)e^(ū) ^(j) as equation (3) might have suggested. Incorporatingthe interaction of quality and price enables us to characterize howquality differences translate to price differences across products,thereby reconciling the counterintuitive equal-markup solution in theliterature.

Let p=(p_(i))_(j∈l) be the vector of prices of the new products. Thefirm's price optimization problem is

${{\max\limits_{p}\; {\pi (p)}} = {{\sum\limits_{i \in I}{\left( {p_{i} - c_{i}} \right){q_{i}(p)}}} + {\sum\limits_{j \in J}{{\overset{\_}{m}}_{j}{q_{j}(p)}}}}},$

which is not a concave or a quasiconcave maximization even for thespecial case of J=∅ (Dong et al., 2009). Profit is rewritten as afunction of choice probabilities of the new products, q=(q_(i))_(i∈l),and show that this function is concave. From (5) and (6),

$\begin{matrix}{{{p_{i} = {{\left( {{\overset{\_}{x}}_{i} + a_{i} + {\log \; q_{0}} - {\log \; q_{i}}} \right)/\left( {b_{i} - {\beta_{i}{\overset{\_}{x}}_{i}}} \right)}\mspace{14mu} {and}}}q_{0} = {{1 - {\sum\limits_{i \in I}q_{i}} - {\sum\limits_{j \in J}q_{j}}} = {1 - {\sum\limits_{i \in I}q_{i}} - {\left( {\sum\limits_{j \in J}e^{{\overset{\_}{u}}_{j}}} \right)q_{0}}}}},} & (7)\end{matrix}$

the latter of which is equivalent to

$\begin{matrix}{q_{0} = {\frac{1 - {\sum\limits_{i \in I}q_{i}}}{1 + {\sum\limits_{j \in J}e^{{\overset{\_}{u}}_{j}}}}.}} & (8)\end{matrix}$

The price optimization problem can be restated as

$\begin{matrix}{\mspace{79mu} {{{{\max\limits_{q}\; {\hat{\pi}(q)}} = {{\sum\limits_{i \in I}{\left( {{p_{i}(q)} - c_{i}} \right)q_{i}}} + {\sum\limits_{j \in J}{{\overset{\_}{m}}_{j}{q_{j}(q)}}}}},\mspace{79mu} {where}}{{p_{i}(q)} = {{\left\lbrack {{\overset{\_}{x}}_{i} + a_{i} + {\log \left( \frac{1 - {\sum\limits_{i \in I}q_{i}}}{1 + {\sum\limits_{j \in J}e^{{\overset{\_}{u}}_{i}}}} \right)} - {\log \; q_{i}}} \right\rbrack/\left( {b_{i} - {\beta_{i}{\overset{\_}{x}}_{i}}} \right)}\mspace{14mu} {and}}}\mspace{85mu} {{q_{j}(q)} = {\frac{1 - {\sum\limits_{i \in I}q_{i}}}{1 + {\sum\limits_{j \in J}e^{{\overset{\_}{u}}_{j}}}}{e^{{\overset{\_}{u}}_{j}}.}}}}} & (9)\end{matrix}$

Theorem 1. {circumflex over (π)}(q) is concave in q.

The proof of Theorem 1 follows the logic presented in the proof ofTheorem 3, and is omitted. Previous works have shown that the profit isconcave in the choice probability vector for a clean-slate problem inwhich J=∅. Theorem 1 extends the state-of-art literature and establishesa unique optimal solution in the presence of existing products.

Theorem 2. The optimal prices of the new products and the firm's optimaltotal profit are

$\begin{matrix}{{p_{i}^{*} = {c_{i} + \frac{1}{b_{i} - {\beta_{i}{\overset{\_}{x}}_{i}}} + \theta}}{\pi^{*} = \theta}} & (10)\end{matrix}$

where θ solves the single-variable equation

$\begin{matrix}{\theta = {{\overset{\_}{\pi}}_{j} + {\frac{\sum\limits_{i \in I}{e^{{\overset{\_}{x}}_{i} + a_{i} - 1 - {{({b_{i} - {\beta_{i}{\overset{\_}{x}}_{i}}})}{({c_{i} + \theta})}}}/\left( {b_{i} - {\beta_{i}{\overset{\_}{x}}_{i}}} \right)}}{1 + {\sum\limits_{j \in J}e^{{\overset{\_}{u}}_{j}}}}.}}} & (11)\end{matrix}$

From Theorem 2, all else equal, the optimal markup is higher forhigher-quality product, and for product with lower b_(i) and higherβ_(i) values, as stated in the following corollary.

Corollary 1. At optimality, the following holds for

i, i′∈I, i≠i′: (i) let b _(i) =b _(p) and β_(i)=β_(p).

Then, p*_(i) c _(i) >p* _(p) −c _(p) if and only if x _(i) >x _(p). (ii)Let β_(i)=β_(p) and x _(i) =x _(p). Then p* _(i) −c _(i) >p* _(p) −c_(p) if and only if b _(i) <b _(p). (iii) Let b _(i) =b _(p) and x _(i)=x _(p). Then p* _(i) −c _(i) >p* _(i′) −c _(i′) if and only ifβ_(i)>β_(p).

Hence, controlling other parameters, a hotel room with a higher resortcredit package or a smartphone with larger storage size should command ahigher markup than its peer products. The next corollary implies thatthe well-known equal mark-up property holds if β_(i)=0 and b_(i)=b forall l∈l.

Corollary 2. If b_(i)=b and β_(i)=0 for all i∈l, then the optimal pricesbecome

$p_{i}^{*} = {c_{i} + {\overset{\_}{\pi}}_{J} + {\frac{1}{b}\left\lbrack {1 + {W\left( \frac{\sum\limits_{i \in I}e^{{\overset{\_}{x}}_{i} + a_{i} - {bc}_{i} - 1 - {b\; {\overset{\_}{\pi}}_{j}}}}{1 + {\sum\limits_{j \in J}e^{{\overset{\_}{u}}_{j}}}} \right)}} \right\rbrack}}$

where W(·) is the Lambert W function.

It should be remarked that, treating b_(i)−β_(i) x _(i) as the effectiveprice sensitivity, the relationship in (10) reproduces the more generalequal “adjusted mark-up” property identified previously, but specializesit in terms of quality-price interaction.

Theorem 2 also leads to the following bounds for π*.

Corollary 3.

${\overset{\_}{\pi}}_{J} \leq \pi^{*} \leq {{\overset{\_}{\pi}}_{J} + {\frac{\sum\limits_{i \in I}{e^{{\overset{\_}{x}}_{i} + a_{i} - 1 - {{({b_{i} - {\beta_{i}{\overset{\_}{x}}_{i}}})}{({c_{i} + {\overset{\_}{\pi}}_{J}})}}}/\left( {b_{i} - {\beta_{i}{\overset{\_}{x}}_{i}}} \right)}}{1 + {\sum\limits_{j \in J}e^{{\overset{\_}{u}}_{j}}}}.}}$

These bounds, along with equation (11), lead to an efficient bisectionsearch algorithm for solving the optimal profit and prices.

Algorithm 1. (Price Optimization)${1.\mspace{14mu} {Let}\mspace{14mu} \theta^{-}} = {{{\overset{\_}{\pi}}_{J}\mspace{14mu} {and}\mspace{14mu} \theta^{+}} = {{\overset{\_}{\pi}}_{J} + {\frac{\sum_{i \in I}e^{{\overset{\_}{x}}_{i} + a_{i} - 1 - {({b_{i} - {\beta_{i}{\overset{\_}{x}}_{i}}})} + {{({c_{i} + {\overset{\_}{\pi}}_{J}})}/{({b_{i} - {\beta_{i}{\overset{\_}{x}}_{i}}})}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}.}}}$2. Let θ = (θ⁻ + θ⁺)/2.${3.\mspace{14mu} {Compute}\mspace{14mu} f} = {{\overset{\_}{\pi}}_{J} + {\frac{\sum_{i \in I}e^{x_{i} + a_{i} - 1 - {({b_{i} - {\beta_{i}{\overset{\_}{x}}_{i}}})} + {{({c_{i} + \theta})}/{({b_{i} - {\beta_{i}{\overset{\_}{x}}_{i}}})}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}.}}$4. If f > θ, let θ⁻ = θ; if f < θ, let θ⁺ = θ. 5. Repeat Steps 2-4 untilf = θ. 6. Compute optimal prices according to equation (10).

EXAMPLES

Consider a manufacturer with a product cost function c(a, x)=0.5a+x².Suppose the manufacturer currently offers three products with a_(j), x_(j) and p _(j) values shown in Table 3. The manufacturer plans to addthree new products, products 4-6, with attributes given in Table 3,while still keeping products 1-3 in its portfolio and maintaining theircurrent prices. The price coefficients are b_(i)=b_(j)=1 andβ_(i)=β_(j)=0.2 for all i∈l and j∈J. Algorithm 1 is applied to obtainthe optimal prices for the new products. It was observed that theoptimal markups vary across products despite that all products have thesame b and β values. This is more in line with practice than theequal-markup solution.

TABLE 3 Price Optimization. Initial New Products Products (j ϵ J) α_(j){tilde over (x)}_(j) {tilde over (p)}_(j) {tilde over (m)}_(j) (i ϵ I)α_(i) {tilde over (x)}_(i) p_(i)* m_(i)* 1 0.0 0.5 2 1.75 4 0.0 0.8 3.392.75 2 1.0 0.8 3 1.86 5 1.0 1.0 4.31 2.81 3 2.0 1.0 4 2.00 6 2.0 1.25.32 2.86

Table 4 presents a comparison of problem instances and sheds light onhow optimal prices are affected by the magnitude of b, β and the qualityof the new products. In these examples, the quality and prices for threeexisting products are {a}_(j∈J)=[0, 1, 2], {{umlaut over(x)}}_(j∈J)=[0.5, 0.8, 1.0] and {{umlaut over (p)}}_(j∈J)=[2, 3, 4]. Thecost function is the same as in the example of Table 3. The comparisonof instances 1-3 demonstrates the effect of the parameter b while thecomparison of instances 2, 4 and 5 demonstrates the effect of theparameter β—the optimal prices decrease in b and increase in β.Instances 2, 6, and 7 suggest that higher quality levels lead to higherprices while instances 7-9 show that a larger quality gap betweenproducts (i.e., larger x _(i)−x _(i′) value) results in a larger pricegap (i.e., larger p_(i) ⁺-p_(p) ⁺ value).

TABLE 4 Optical Quality Vary with Prices. New Quality Optimal PricesInstance b β {tilde over (x)}₄ {tilde over (x)}₅ {tilde over (x)}₆ p₄*p₅* p₆* profit 1 2.00 0.05 0.8 1.0 1.2 1.39 2.25 3.19 0.24 2 1.00 0.050.8 1.0 1.2 2.95 3.83 4.78 1.27 3 0.50 0.05 0.8 1.0 1.2 5.09 6.00 6.992.27 4 1.00 0.10 0.8 1.0 1.2 3.09 3.98 4.94 1.36 5 1.00 0.20 0.8 1.0 1.23.39 4.31 5.32 1.56 6 1.00 0.05 0.3 0.5 0.7 2.42 3.09 3.85 1.32 7 1.000.05 0.2 0.4 0.6 2.36 3.00 3.71 1.31 8 1.00 0.05 0.2 0.5 0.8 2.36 3.093.99 1.31 9 1.00 0.05 0.5 1.0 1.5 2.51 3.79 5.57 1.24

Attribute Optimization

Next, the attribute optimization problem will be presented in which thefirm optimizes the attribute values x_(i) of the new products i∈l whilethe attribute values of existing product j∈J are fixed at x _(j).Product prices are exogenously given, denoted by p _(i), i∈I∪J. Forexample, the resort hotel in Table 1 may create three new room offeringsat the price levels $289, $359 and $429 respectively and wish tooptimize the service package value for each new room offering. In thehotel example, a_(i) represents customers' utility for a particular roomtype (i.e., King, Queen or Suite) and x_(i) represents customers'utility for a particular service package (e.g., $50, $80, or $100 resortcredit). In general, the value of a_(i) reflects the composite utilityof all attributes of product i that are not part of the design decision(i.e., features of the product that are not to be changed), whereasx_(i) is the utility from the attribute to be optimized. The choice ofattribute value affects product cost. Let c_(i)(x_(i)) be the unit costof product i∈l at attribute value x_(i), which is assumed to benonnegative and strictly increasing. The product cost function maydiffer across products, reflecting differences in fixed and variableattributes, i.e., c_(i)(x_(i))=c(a_(i), x_(i)). For brevity, we denotethe cost function with c_(i)(x_(i)) but we emphasize that it is also afunction of a_(i).

Let x=(x_(i))_(i∈l) denote the vector of attribute values for the newproducts. The attribute optimization problem is

${\max\limits_{x}{\pi (x)}} = {{\sum\limits_{i \in I}{\left( {{\overset{\_}{p}}_{i} - {c_{i}\left( x_{i} \right)}} \right){q_{i}(x)}}} + {\sum\limits_{j \in J}{{\overset{\_}{m}}_{j}{q_{j}(x)}}}}$

where m _(j) is the contribution margin of product j∈J.

Observe that setting x_(i)=b_(i)/β_(i) for all i∈l yields infiniteoptimal profit if c_(i)(b_(i)/β_(i)) is finite for all i∈l, i.e.,

${\pi \left( {\left( {b_{i}/\beta_{i}} \right)_{i \in I},P} \right)} = {{\sum\limits_{i \in I}{\left( {p_{i} - {c_{i}\left( {b_{i}/\beta_{i}} \right)}} \right)\left( \frac{e^{b_{i}/\beta_{i}}}{1 + {\sum\limits_{k \in I}e^{b_{k}/\beta_{k}}} + {\sum\limits_{j \in J}e^{{\overset{\_}{u}}_{j}}}} \right)}} + {\sum\limits_{j \in J}{{\overset{\_}{m}}_{j}\left( \frac{e^{{\overset{\_}{u}}_{j}}}{1 + {\sum\limits_{k \in I}e^{b_{k}/\beta_{k}}} + {\sum\limits_{j \in J}e^{{\overset{\_}{u}}_{j}}}} \right)}}}$

which goes to infinity as p→∞. To avoid this pathological solution, weassume that the cost function becomes unbounded prior to x=min_(i∈l)b_(i)/β_(i), i.e., technological constraints are such that it isimpossible to produce a product with non-price attribute of x=min_(i∈l)b_(i)/β_(i). We let x⁺<min_(i∈l) b_(i)/β_(i) denote the technologicalupper limit of quality, i.e.,

From (5),

$\begin{matrix}{x_{i} = {\frac{{b_{i}{\overset{\_}{p}}_{i}} + {\log \; q_{i}} - {\log \; q_{0}}}{1 + {\beta_{i}{\overset{\_}{p}}_{i}}} - {a_{i}.}}} & (12)\end{matrix}$

From (8), q₀ is a linear function of q. Substitute (6) and (12) into thetotal profit to obtain the total profit as a function of q.

$\begin{matrix}\begin{matrix}{{\hat{\pi}(q)} = {{\sum\limits_{i \in I}{\left( {{\overset{\_}{p}}_{i} - {c_{i}\left( \frac{{b_{i}{\overset{\_}{p}}_{i}} + {\log \; q_{i}} - {\log \; {q_{0}(q)}}}{1 + {\beta_{i}{\overset{\_}{p}}_{i}}} \right)} - a_{i}} \right)q_{i}}} + {\left( {\sum\limits_{j \in J}{{\overset{\_}{m}}_{j}e^{{\overset{\_}{u}}_{j}}}} \right){q_{0}(q)}}}} \\{= {{\sum\limits_{i \in I}{{\overset{\_}{p}}_{i}q_{i}}} - {\sum\limits_{i \in I}{{c_{i}\left( {\frac{{b_{i}{\overset{\_}{p}}_{i}} + {\log \; q_{i}} - {\log \; {q_{0}(q)}}}{1 + {\beta_{i}{\overset{\_}{p}}_{i}}} - a_{i}} \right)}q_{i}}} +}} \\{{\left( {\sum\limits_{j \in J}{{\overset{\_}{m}}_{j}e^{{\overset{\_}{u}}_{j}}}} \right){{q_{0}(q)}.}}}\end{matrix} & (13)\end{matrix}$

The first and third terms are both linear in q (see equation (8)), thusif the term

${c_{i}\left( {\frac{{b_{i}{\overset{\_}{p}}_{i}} + {\log \; q_{i}} - {\log \; {q_{0}(q)}}}{1 + {\beta_{i}{\overset{\_}{p}}_{i}}} - a_{i}} \right)}q_{i}$

is convex in q, then the total profit is concave in q. To establishconvexity of this cost term, we make use of Lemma 1 which is ageneralization of Lemma 2.Lemma 1. Let φ(z₁, z₂)=z₁f (k[log z₁-log(1−z₂)]+δ) where k>0 and δ areconstants. Assume f′(·)+k f″(·)≥0. Then φ is jointly convex on [0,1]².Assumption 1. The cost function c_(i)(x_(i)), i∈l is twicedifferentiable and satisfies

$\begin{matrix}{{{c_{i}^{\prime}\left( x_{i} \right)} + {\frac{1}{1 + {\beta_{i}{\overset{\_}{p}}_{i}}}{c_{i}^{''}\left( x_{i} \right)}}} > {0\mspace{14mu} {for}\mspace{14mu} {all}\mspace{14mu} x_{i}} \leq {x_{i}^{+}.}} & (14)\end{matrix}$

Assumption 1 ensures that the cost function is well-behaved and issatisfied by any increasing and convex cost function.

Theorem 3. Suppose Assumption 1 holds. Then {circumflex over (π)}(q) isconcave in q.

Since the total profit is concave in the choice probability vector q,the optimal solution is unique and can be derived from the first-ordercondition. Take the derivative of {circumflex over (π)}(q) with respectto q_(i) and set it to zero to obtain

$\begin{matrix}{{{\overset{\_}{p}}_{i} - {c_{i}\left( {\frac{{b_{i}{\overset{\_}{p}}_{i}} + {\log \; q_{i}} - {\log \; q_{0}}}{1 + {\beta_{i}{\overset{\_}{p}}_{i}}} - a_{i}} \right)} - {\frac{1}{1 + {\beta_{i}{\overset{\_}{p}}_{i}}}{c_{i}^{\prime}\left( {\frac{{b_{i}{\overset{\_}{p}}_{i}} + {\log \; q_{i}} - {\log \; q_{0}}}{1 + {\beta_{i}{\overset{\_}{p}}_{i}}} - a_{i}} \right)}}} = {\frac{\sum_{j \in J}{{\overset{\_}{m}}_{j}e^{{\overset{\_}{u}}_{j}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}} + {\frac{\sum_{i^{\prime} \in I}{\frac{1}{1 + {\beta_{i^{\prime}}{\overset{\_}{p}}_{i^{\prime}}}}q_{i}{c_{i^{\prime}}^{\prime}\left( {\frac{{b_{i^{\prime}}{\overset{\_}{p}}_{i^{\prime}}} + {\log \; q_{i^{\prime}}} - {\log \; q_{0}}}{1 + {\beta_{i^{\prime}}{\overset{\_}{p}}_{i^{\prime}}}} - a_{i^{\prime}}} \right)}}}{1 - {\sum_{i^{\prime} \in I}q_{i^{\prime}}}}.}}} & (15)\end{matrix}$

Note that the right side of (15) is independent of i. Substituting (3),(8) and (12), we can rewrite the first order condition as

$\begin{matrix}{{{\overset{\_}{p}}_{i} - {c_{i}\left( x_{i} \right)} - {\frac{1}{1 + {\beta_{i}{\overset{\_}{p}}_{i}}}{c_{i}^{\prime}\left( x_{i} \right)}}} = {\frac{\sum_{j \in J}{{\overset{\_}{m}}_{j}e^{{\overset{\_}{u}}_{j}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}} + {\frac{\sum_{i^{\prime} \in I}{\frac{1}{1 + {\beta_{i^{\prime}}{\overset{\_}{p}}_{i^{\prime}}}}e^{x_{i^{\prime}} + a_{i^{\prime}} - {b_{i^{\prime}}{\overset{\_}{p}}_{i^{\prime}}} + {\beta_{i^{\prime}}x_{i^{\prime}}{\overset{\_}{p}}_{i^{\prime}}}}{c_{i^{\prime}}^{\prime}\left( x_{i^{\prime}} \right)}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}.}}} & (16)\end{matrix}$

For any given θ, let x(θ) denote the vector of attribute values thatsolves

$\begin{matrix}{{{{\overset{\_}{p}}_{i} - {c_{i}\left( x_{i} \right)} - {\frac{1}{1 + {\beta_{i}{\overset{\_}{p}}_{i}}}{c_{i}^{\prime}\left( x_{i} \right)}}} = \theta},{i \in {I.}}} & (17)\end{matrix}$

By Assumption 1, the left side of (17) decreases in x_(i). Thus for anygiven θ, the solution of Equation (17) is unique and x(θ) is decreasingin θ.

Lemma 2: Suppose Assumption 1 holds. Then for any given θ, x (θ) isunique.

Therefore, we can compare the optimal attribute values across productsutilizing equation (17).

Corollary 4. For any i, i′ ∈l and i≠i′,

let β_(i)=β_(i′) and c _(i)(·)=c _(i′)(·), then x _(i) ^(k) >x _(i′)^(k) if and only if p _(i) >p _(i′).  (i)

let p _(i) =p _(i′) and c _(i)(·)=c _(i′)(·), then x _(i) ^(k) >x _(i′)^(k) if and only if β_(i)>β_(i′).  (ii)

The sign of x_(i) ^(k)−x_(i′) ^(k) is independent of b_(i) and b_(i′)values.  (iii)

Result (i) states that, all else equal, a higher-priced product shouldbe matched with a higher attribute, which is consistent withobservations in practice. Since a larger interaction coefficient β_(i)implies that customers are more sensitive to quality increment for thisproduct, result (ii) is also expected. Result (iii) indicates that themagnitude of b_(i) does not affect how the optimal attribute valuescompare across products; this result draws an interesting contrast withthe joint price and attribute optimization discussed later herein.

From Lemma 2, the total profit can be expressed as a function of θ:

${\overset{\sim}{\pi}(\theta)} = {{\sum\limits_{i \in I}{\left( {p_{i} - {c_{i}\left( {x_{i}(\theta)} \right)}} \right){q_{i}\left( {x(\theta)} \right)}}} + {\sum\limits_{j \in J}{{\overset{\_}{m}}_{j}{{q_{j}\left( {x(\theta)} \right)}.}}}}$

The next theorem shows that a fixed-point relationship holds atoptimality.

Theorem 4. Let θ*=argmax_(θ){tilde over (π)}(θ), Then ϑ satisfies {tildeover (π)}(θ)=θ and is the unique solution of the equation

$\begin{matrix}{\theta^{*} = {\frac{\sum_{j \in J}{{\overset{\_}{m}}_{j}e^{{\overset{\_}{u}}_{j}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}} + {\frac{\sum_{i \in I}{\frac{1}{{1 + {\beta_{i}{\overset{\_}{p}}_{i}}}\;}e^{{x_{i}{(\theta^{*})}} + a_{i} - {b_{i}{\overset{\_}{p}}_{i}} + {\beta_{i}{x_{i}{(\theta^{*})}}{\overset{\_}{p}}_{i}}}{c_{i}^{\prime}\left( {x_{i}\left( \theta^{*} \right)} \right)}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}.}}} & (18)\end{matrix}$

Next, how to find the optimal solution will be considered. It can beshown that the right side of (18) decreases in θ* for cost functionssatisfying (14) (see the proof of Theorem 4). The left side increases inθ*. Therefore, the above equation can be efficiently solved with abisection search. From equations (17) and (18), and the fact that theright side of (18) is decreasing in θ*, θ* is bounded in the followinginterval.

Corollary 5.

${\overset{\_}{\pi}}_{J} \leq \theta^{*} \leq {\min {\left\{ {\frac{\sum_{i \in I}{\frac{1}{1 + {\beta_{i}p_{i}}}e^{{x_{i}{({\overset{\_}{\pi}}_{J})}} + a_{i} - {b_{i}{\overset{\_}{p}}_{i}} + {\beta_{i}{x_{i}{({\overset{\_}{\pi}}_{J})}}{\overset{\_}{p}}_{i}}}{c_{i}^{\prime}\left( {x_{i}\left( {\overset{\_}{\pi}}_{J} \right)} \right)}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}},{\min_{i \in I}{\overset{\_}{p}}_{i}}} \right\}.}}$

Therefore, for an interior solution θ* to exist, the following conditionis needed.

Assumption 2. {umlaut over (π)}_(J)<{umlaut over (p)}_(i) for all i∈l.

If Assumption 2 does not hold, then at least for some i (those withlowest price), the first-order derivative

$\frac{\partial\hat{\pi}}{\partial q_{i}}$

is always negative. Therefore, the product with the lowest price shouldhave its quantity (and thus attribute) set to as low as possible. Moregenerally, a sufficient condition for

$\frac{\partial{\hat{\pi}(q)}}{\partial q_{i}} < 0$

for any q is p_(i)≤{umlaut over (π)}_(J) which implies product i shouldnot be included in the portfolio. This is consistent with the knownresult on assortment under MNL. Talluri and van Ryzin (2004) show thatthe optimal assortment is a revenue-ordered assortment consisting ofproducts with the highest revenues. We remark that such apre-optimization condition does not exist for the price optimizationproblem discussed herein, which is also consistent with the known resulton joint price and assortment optimization under MNL that the optimalassortment is to include all products.

From (18) and Corollary 5, the following algorithm is derived forattribute optimization.

Algorithm 2. (Attribute Optimization)${1.\mspace{14mu} {Let}\mspace{14mu} \theta^{-}} = {{{\overset{\_}{\pi}}_{J}\mspace{14mu} {and}\mspace{14mu} \theta^{+}} = {\min {\left\{ {\frac{\sum_{i \in I}{\frac{1}{1 + {\beta_{i}\overset{\_}{p_{i}}}}e^{{x_{i}{({\overset{\_}{\pi}}_{J})}} + a_{i} - {b_{i}{\overset{\_}{p}}_{i}} + {\beta_{i}{{\overset{\_}{x}}_{i}{({\overset{\_}{\pi}}_{J})}}{\overset{\_}{p}}_{i_{c_{i}^{\prime}{({x_{i}{({\overset{\_}{\pi}}_{J})}})}}}}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}},{\min_{i \in I}{\overset{\_}{p}}_{i}}} \right\}.}}}$2. Let θ = (θ⁻ + θ⁺)/2. 3. Solve Equation (17) through the followingbisection search to obtain x_(i)(θ) for all i ∈ I:  (a) Let y⁻ = 0 andy⁺ = x_(i) ⁺.  (b) Let y = (y⁻ + y⁺)/2.   ${(c)\mspace{14mu} {Compute}\mspace{14mu} g} = {{\overset{\_}{p}}_{i} - {c_{i}(y)} - {\frac{1}{1 + {\beta_{i}{\overset{\_}{p}}_{i}}}{{c_{i}^{\prime}(y)}.}}}$ (d) if g > θ, let y⁻ = y; if g < θ, let y⁺ = y.  (e)Repeat Steps(a)-(e) until g = θ. Then x_(i)(θ) = y.${4.\mspace{14mu} {Compute}\mspace{14mu} f} = {\frac{\sum_{j \in J}{{\overset{\_}{m}}_{j}e^{{\overset{\_}{u}}_{j}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}} + {\frac{\sum_{i \in I}{\frac{1}{1 + {\beta_{i}{\overset{\_}{p}}_{i}}}e^{{x_{i}{(\theta)}} + a_{i} - {b_{i}{\overset{\_}{p}}_{i}} + {\beta_{i}{x_{i}{(\theta)}}{\overset{\_}{p}}_{i_{c_{i}^{\prime}{({x_{i}{(\theta)}})}}}}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}.}}$5. If f > θ, let θ⁻ = θ; if f < θ, let θ⁺ = θ. 6. Repeat Steps 2-5 untilf = θ. The optimal attribute values are given by x_(i)(θ) obtained inStep 3.

EXAMPLES

Consider a manufacturer with a product cost function c(a,x)=0.1a+1.5e^(1.5x). The manufacturer currently offers three products inits portfolio, products 1-3, with {a}_(j∈J)=[6, 9, 12], {x}_(j∈J)=[0.6,0.8, 1.0] and {p}_(j∈J)=[7, 8, 10]. The manufacturer plans to introducethree new products, products 4-6, with attributes {a},_(j∈l)=[6, 9, 12]while still keeping products 1-3 in the product portfolio. The newproduct prices p _(i) are given in Table 5. Algorithm 2 is applied tooptimize attributes x_(i), i∈l of the new products.

TABLE 5 Optimizing Attributes of New Products New Prices OptimalAttributes Instance b β {tilde over (p)}₄ {tilde over (p)}₅ {tilde over(p)}₆ x₄* x₅* x₆* profit 1 1.0 0.30 6.3 7.2 9 0.18 0.32 0.59 2.71 2 1.00.30 7.7 8.8 11 0.49 0.62 0.85 2.56 3 1.0 0.35 6.3 7.2 9 0.22 0.36 0.622.62 4 1.0 0.35 7.7 8.8 11 0.51 0.64 0.87 2.55 5 1.0 0.40 6.3 7.2 9 0.260.39 0.65 2.54 6 1.0 0.40 7.7 8.8 11 0.53 0.66 0.89 2.55 7 1.5 0.30 6.37.2 9 0.22 0.36 0.61 2.51 8 1.5 0.30 7.7 8.8 11 0.57 0.69 0.90 1.96 91.5 0.35 6.3 7.2 9 0.24 0.38 0.63 2.54 10 1.5 0.35 7.7 8.8 11 0.58 0.690.91 2.10 11 1.5 0.40 6.3 7.2 9 0.26 0.39 0.65 2.54 12 1.5 0.40 7.7 8.811 0.58 0.70 0.92 2.20 13 2.0 0.30 6.3 7.2 9 0.58 0.67 0.85 0.27 14 2.00.30 7.7 8.8 11 0.78 0.87 1.05 0.09 15 2.0 0.35 6.3 7.2 9 0.59 0.68 0.860.37 16 2.0 0.35 7.7 8.8 11 0.80 0.89 1.06 0.14 17 2.0 0.40 6.3 7.2 90.59 0.68 0.87 0.49 18 2.0 0.40 7.7 8.8 11 0.81 0.90 1.07 0.21

Observations of the attribute optimization results in Table 5 indicatethat high-priced products are matched with high attribute values. Inaddition, the optimal attributes increase with price sensitivityparameter b. While the limited instances in Table 5 seem to indicatethat the optimal attributes also increase with β, more extensivenumerical experiments, however, show that this trend is not always anincreasing one, nor is it necessarily monotonic. FIGS. 1A and 1Billustrate these trends in more detail.

The trend that optimal attributes increase with price sensitivity (FIG.1A appears counter-intuitive: with higher price sensitivity, customersare less willing to pay high prices; thus it would seem wise not toincrease product attributes. However, this dynamic is absent in theattribute optimization problem because prices of the new products arefixed; therefore, as customers become more price sensitive, themanufacturer has to attract them with higher attributes to compete withthe no-purchase option. Therefore, the increasing trend in FIG. 1Aresults from the substitution of attribute for price. As we willdemonstrate later, this trend is reversed when attributes and prices areoptimized jointly.

The non-monotonic pattern in FIG. 1B suggests that the impact of β onthe attribute decision is multifaceted. In particular, recall that the βparameter is the coefficient of the interaction of attribute and priceand that it affects the marginal utility of attribute x (see equation(2)). As β increases, customers obtain higher marginal utility of x,creating an incentive for the manufacturer to raise x. On the flip side,as β further increases, the products become more and more competitiverelative to no-purchase option and there is diminishing return togaining additional market share by increasing attribute. And since themanufacturer cannot raise price, the optimal strategy is to shift fromprioritizing gains in market share to prioritizing reductions in cost byreducing x values.

Attribute and Price Optimization

In this section, the joint optimization of product attribute and priceis disclosed. For example, the resort hotel considers optimizing boththe service package value and the price of each new room offer tomaximize the total profit. Let x=(x_(i))_(i∈l) denote the vector ofquality levels for the new products and define Ω={x|0≤x_(i)≤x_(i) ⁺,i∈I}.

${\max\limits_{x,p}{\pi \left( {x,p} \right)}} = {{\sum\limits_{i \in I}{\left( {p_{i} - {c_{i}\left( x_{i} \right)}} \right){q_{i}\left( {x,p} \right)}}} + {\sum\limits_{j \in J}{{\overset{\_}{m}}_{j}{{q_{j}\left( {x,p} \right)}.}}}}$

Contrasting this with attribute optimization, which prescribes theoptimal attribute values for the new products that differ acrossproducts due to differences in price p _(i) and the exogenous attributea_(i), the resulting differences in optimal prices and optimalattributes among the new products under joint optimization are solelydue to differences in a_(i). If we interpret the a_(i) value asrepresenting a certain product type, for example, room type in a hotel,type of seats in an airplane (main cabin, business class, first class),or size of a rental car (compact, mid-size, full-size) and the x_(i)value as representing add-on services, the joint attribute-priceoptimization model helps us optimally set both the level of add-onservices and the price for the product to be offered in each type.

Recall that in the hotel example, a_(i) represents customers' utilityfor a particular room type (i.e., King, Queen or Suite) and x_(i)represents customers' utility for a particular service package (e.g.,$50, $80, or $100 resort credit). In general, the value of a_(i)reflects the composite utility of all attributes of product i that arenot part of the design decision (i.e., features of the product that arenot to be changed), whereas quality x_(i) is the utility from theattribute to be optimized. For example, a_(i) may reflect a certainproduct type, for example, room type in a hotel, type of seats in anairplane (main cabin, business class, first class), or size of a rentalcar (compact, mid-size, full-size) and x_(i) may represent add-onservices. The joint quality-price optimization model helps to optimallyset both the level of add-on services and the price for the product tobe offered in each type.

The choice of quality affects product cost. Let c_(i)(x_(i)) be the unitcost of product i∈I at quality x_(i), which is assumed to benonnegative. The product cost function may differ across products,reflecting differences in fixed and variable attributes, i.e.,c_(i)(x_(i))=c(a_(i), x_(i)). For brevity, we denote the cost functionwith c_(i)(x_(i)) but we emphasize that it is also a function of a_(i).To ensure that the cost function c_(i)(x_(i)) is well-behaved, we makethe following assumption.

Assumption 3. The cost function c_(i)(x_(i)), i∈I is twicedifferentiable, increasing and convex in x_(i) for all x_(i)∈[0, x_(i)⁺].

It should be noted that the convexity of c_(i)(x_(i)) does notnecessitate convexity of the nominal cost curve of a product attribute.Note that x_(i) is a linear utility measure of quality that can bedifferent from its natural or nominal measure. In the smart phoneexample, suppose that the cost of memory increases linearly with size.Since the x values are generated by taking logarithm, the cost functionin terms of x becomes exponential which is convexly increasing. Asimilar argument holds for the hotel example. Suppose customer utilitydoes not grow linearly with the resort credit amount, but at a lowerorder of growth (e.g., the rate of a square root function) while thecost of the resort credit grows linearly with the amount. Then the costfunction in terms of x becomes convex (e.g., quadratic). In other words,Assumption 3 is satisfied if the cost of the focal attribute is convexlyincreasing in its linear utility contribution x_(i) or equivalently, theutility of the focal attribute exhibits diminishing return on cost.

Let m=(m _(i))_(i∈I) where m _(i) =p _(i) −c _(i)(x _(i)). From (5), q_(i) =q ₀ e ^(x) ^(i) ^(+a) ^(i) ^(−b) ^(i) ^(p) ^(i) ^(+β) ^(i) ^(x)^(i) ^(p) ^(i) =q ₀ e ^(x) ^(i) ^(+a) ^(i) ^(−(b) ^(i) ^(−β) ^(i) ^(x)^(i) ^()p) ^(i) =q ₀ e ^(x) ^(i) ^(+a) ^(i) ^(−(b) ^(i) ^(−β) ^(i) ^(x)^(i) ^()c) ^(i) ^(−(b) ^(i) ^(−β) ^(i) ^(x) ^(i) ^()m) ^(i)

and thus solve m_(i) as a function of x and q:

m _(i) =|x _(i) +a _(i)−(b _(i)−β_(i) x _(i))c _(i)(x _(i))−log q_(i)+log q ₀|/(b _(i)−β_(i) x _(i)).

We can express the total profit as

${\hat{\pi}\left( {x,q} \right)} = {{\sum\limits_{i \in I}{\frac{q_{i}}{b_{i} - {\beta_{i}x_{i}}}\left\lbrack {x_{i} + a_{i} - {\left( {b_{i} - {\beta_{i}x_{i}}} \right){c_{i}\left( x_{i} \right)}} - {\log \; q_{i}} + {\log \; {q_{0}(q)}}} \right\rbrack}} + {\left( {\sum\limits_{j \in J}{{\overset{\_}{m}}_{j}e^{{\overset{\_}{u}}_{j}}}} \right){{q_{0}(q)}.}}}$

Theorem 5. Given {circumflex over (π)}(x,q) is concave in the choiceprobability vector q. The optimal markup is given by

${m_{i}^{*}(x)} = {\frac{1}{b_{i} - {\beta_{i}x_{i}}} + {\theta (x)}}$

where θ(x) solves

$\begin{matrix}{\theta = {\frac{\sum_{i \in I}{e^{x_{i} + a_{i} - {{({b_{i} - {\beta_{i}x_{i}}})}{c_{i}{(x_{i})}}} - 1 - {{({b_{i} - {\beta_{i}x_{i}}})}\theta}}/\left( {b_{i} - {\beta_{i}x_{i}}} \right)}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}} + {\frac{\sum_{j \in J}{{\overset{\_}{m}}_{j}e^{{\overset{\_}{u}}_{j}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}.}}} & (19)\end{matrix}$

Let {tilde over (π)}(x)=max_(q) {circumflex over (π)}(x, q). Thefollowing relationship holds.

Lemma 3. {tilde over (π)}(x)=θ(x).

Therefore, to maximize {tilde over (π)}(x), only θ(x) needs to bemaximized.

In the special case when β_(i)=0, the optimal solution is unique andgiven in the following theorem.

Theorem 6 Let x⁺=(x*_(i))_(i∈I) and p⁺=(p*_(i))_(i∈I) be the optimalsolution to the joint quality and price optimization problem. Supposeβ_(i)=0. Then the optimal solution is given by

$\begin{matrix}{x_{i}^{*} = \left\{ {\begin{matrix}{{c_{i}^{\prime - 1}\left( \frac{1}{b_{i}} \right)},} & {{{if}\mspace{14mu} {c_{i}^{\prime - 1}\left( \frac{1}{b_{i}} \right)}} \in \left\lbrack {0,x_{i}^{+}} \right\rbrack} \\{0,} & {{{if}\mspace{14mu} {c_{i}^{\prime - 1}\left( \frac{1}{b_{i}} \right)}} < 0} \\{x_{i}^{+},} & {{{if}\mspace{14mu} {c_{i}^{\prime - 1}\left( \frac{1}{b_{i}} \right)}} > x_{i}^{+}}\end{matrix},{i \in {I.}}} \right.} & (20) \\{{p_{i}^{*} = {{c_{i}\left( x_{i}^{*} \right)} + \frac{1}{b_{i}} + \theta^{*}}}{{where}\mspace{14mu} \theta^{*}\mspace{14mu} {solves}}\theta = {\frac{\sum_{i \in I}{e^{x_{i}^{*} + a_{i} - {b_{i}{c_{i}{(x_{i}^{*})}}} - 1 - {b_{i}\theta}}/b_{i}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}} + {{\overset{\_}{\pi}}_{J}.}}} & (21)\end{matrix}$

Theorem 6 describes the optimal solution of the joint attribute andprice optimization in the absence of interaction. Consider a costfunction c_(i)(·) that is additively separable in a_(i) and x_(i). Also,consider a case with symmetric b_(i), i.e., b_(i)=b for all i∈l. Fromequations (20) and (21), it must be that x*_(i)=x*_(i′) andp*_(i)−c_(i)(x*_(i))=p*_(i′)−c_(i)(x*_(i′)) for all i, i′∈l. That is,the optimal prices and attributes are such that all new products haveequal markup and equal attribute, as summarized in the followingcorollary.

Corollary 6. Suppose that the cost function is additively separable ina_(i) and x_(i), i.e., c_(i)(x_(i))=c_(a)(a_(i))+c_(x)(x_(i)) and thatb_(i)=b and βi=0 for all i∈l. Then at optimality, x*_(i)=x*_(i), andm*_(i)=m*_(i′) for any i, i′∈l.

The result in Corollary 6 lacks realism and is an oversimplification ofthe effect of attribute and price on customers' utility; however, itserves as a benchmark case for understanding the impact of interaction.Next, we illustrate how the inclusion of a simple attribute and priceinteraction term leads to a different conclusion by capturing a morerealistic relationship between customer preference and productattribute/price.

In general, β_(i)>0 and θ(x) is defined by the implicit function (19).Taking derivatives with respect to x_(i), and with algebraictransformation, we obtain

$\begin{matrix}{\frac{\partial{\theta (x)}}{\partial x_{i}} = {{\frac{q_{i}}{b_{i} - {\beta_{i}x_{i}}}\left\lbrack {1 - {\left( {b_{i} - {\beta_{i}x_{i}}} \right){c_{i}^{\prime}\left( x_{i} \right)}} + {\beta_{i}\theta} + \frac{\beta_{i}}{b_{i} - {\beta_{i}x_{i}}}} \right\rbrack}.}} & (22)\end{matrix}$

Define c_(i) ⁻=c_(i)(0), which is nonnegative, i.e., the cost to producea product at its lowest possible x_(i) value is nonnegative, c_(i) ⁻≥0.Note that

$\frac{\partial{\theta (0)}}{\partial x_{i}} = {\left( \frac{q_{i}}{b_{i}} \right){\left( {1 - {b_{i}{c^{\prime}(0)}} + {\beta_{i}\left( {c^{-} + {\theta (0)} + \frac{1}{b_{i}}} \right)}} \right).}}$

Assumption 4.

$\frac{\partial{\theta (0)}}{\partial x_{i}} > {0\mspace{14mu} {for}\mspace{14mu} {all}\mspace{14mu} {i.}}$

Assumption 4 recites that when the quality of each product is set to thelowest level 0, the profit increases if the quality of any productincreases. This assumption essentially places restrictions on the valuesof b_(i)'s and β_(i)'s.

Lemma 4. If Assumption 4 holds, then x*_(i)∈(0, x⁺)

From Equation (22) and Lemma 4, a necessary condition for optimality is

$\begin{matrix}{{{h_{i}(x)}:={{{- 1} - {\beta_{i}\left( {{\theta (x)} + {c_{i}\left( x_{i} \right)}} \right)} + {\left( {b_{i} - {\beta_{i}x_{i}}} \right){c_{i}^{\prime}\left( x_{i} \right)}} - \frac{\beta_{i}}{b_{i} - {\beta_{i}x_{i}}}} = 0}}\mspace{20mu} {{{{for}\mspace{14mu} {all}\mspace{14mu} i} \in I},}} & (23)\end{matrix}$

which can be rewritten as

$\begin{matrix}{{{{\left( {b_{i} - {\beta_{i}x_{i}}} \right){c_{i}^{\prime}\left( x_{i} \right)}} - {\beta_{i}\left( {{c_{i}\left( x_{i} \right)} + \frac{1}{b_{i} - {\beta_{i}x_{i}}}} \right)}} = {1 + {\beta_{i}\theta}}}{{{for}\mspace{14mu} {all}\mspace{14mu} i} \in I}} & (24)\end{matrix}$

If for any given θ, there exists a unique xi such that the above issatisfied, then the joint attribute and price optimization is reduced toa single-variable fixed point solution. If, in addition, the Jacobian ofh(x)=(h₁(x), h₂(x), . . . , h_(n)(x)) evaluated at x*

${J\left( x^{*} \right)} = \begin{bmatrix}\frac{\partial{h_{1}\left( x^{*} \right)}}{\partial x_{1}} & \ldots & \frac{\partial{h_{1}\left( x^{*} \right)}}{\partial x_{n}} \\\vdots & \ddots & \vdots \\\frac{\partial{h_{n}\left( x^{*} \right)}}{\partial x_{1}} & \ldots & \frac{\partial{h_{n}\left( x^{*} \right)}}{\partial x_{n}}\end{bmatrix}$

is positive semidefinite for any x* satisfying Equation (23), then x* isa global maximum.

In the following theorem, a sufficient condition is identified forpositive semidefinite J(x*) that uses a lower bound on the value

$\frac{c_{i}^{''}(x)}{c_{i}^{\prime}(x)}$

which is a measure of the normalized convexity of the cost functionc_(i)(·). The value of

$\frac{c_{i}^{''}(x)}{c_{i}^{\prime}(x)}$

is generally not difficult to evaluate. For example, for polynomial costfunctions of the form c_(i)(x)=a+bx^(n) where n>1,

${\frac{c_{i}^{''}\left( x_{i} \right)}{c_{i}^{\prime}\left( x_{i} \right)} = \frac{n - 1}{x}};$

for exponential cost functions of the form

Assumption For any x_(i)∈[0, x_(i) ⁺], the cost function c_(i)(·)satisfies

$\begin{matrix}{\frac{c_{i\;}^{''}(x)}{c_{i}^{\prime}\left( x_{i} \right)} > {\frac{3\beta_{i}}{b_{i} - {\beta_{i}x_{i}}}.}} & 25\end{matrix}$

Assumption 5 ensures that for a given θ, the left side of (17) can onlycross 1+β_(i)θ from below. Therefore, if a solution to (17) exists, itmust be unique. Under Assumption 2, the Jacobian matrix J(x*) is adiagonal matrix with nonnegative diagonal elements (note that

$\left. {{\frac{\partial{h_{i}(x)}}{\partial x_{i}} = {{{\left( {b_{i} - {\beta_{i}x_{i}^{*}}} \right){c_{i}^{''}\left( x_{i}^{*} \right)}} - {2\beta_{i}{c_{i}^{\prime}\left( x_{i}^{*} \right)}} - \left( \frac{\beta_{i}}{b_{i} - {\beta_{i}x_{i}^{*}}} \right)^{2}} > {0\mspace{14mu} {and}}}}{\frac{\partial{h_{i}(x)}}{\partial x_{j}} = 0}} \right),$

which is positive semi-definite. This implies global optimality.

Assumption 5 requires that the cost function be “sufficiently” convex.In most realistic scenarios, the interaction effect is small relative tothe main effect of price and it can be expected for the fraction

$\frac{\beta_{i}}{b_{i} - {\beta_{i}x_{i}}}$

to be small. Thus the condition is not as restrictive as it mightappear. For polynomial cost functions of the form ci(x)=a+px_(i′) it canbe shown that the condition reduces to

${\frac{n - 1}{3} > \frac{\beta_{i}x_{i}^{+}}{b_{i} - {\beta_{i}x_{i}^{+}}}};$

for exponential cost functions of the form c_(i)(x)=a+be^(ax), condition(25) reduces to

$\frac{\alpha}{3} > {\frac{\beta_{i}}{b_{i} - {\beta_{i}x_{i}^{+}}}.}$

Theorem 7. If Assumption 5 holds, then the optimal profit θ* to thejoint quality and price optimization problem is the fixed-point solutionto

$\begin{matrix}{\theta = {\frac{\sum_{i \in I}{e^{{x_{i}{(\theta)}} + a_{i} - {{({b_{i} - {\beta_{i}{x_{i}{(\theta)}}}})}{c_{i}{({x_{i}{(\theta)}})}}} - 1 - {{({b_{i} - {\beta_{i}{x_{i}{(\theta)}}}})}\theta}}/\left( {b_{i} - {\beta_{i}{x_{i}(\theta)}}} \right)}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}} + {{\overset{\_}{\pi}}_{J}.}}} & 26\end{matrix}$

where x_(k)(θ) is the unique solution of (17) for any given θ if asolution to (17) exists and x_(i)(θ)=0 or x_(i) ⁺ otherwise(specifically, if

${{1 - {b_{i}{c_{i}^{\prime}(0)}} + {\beta_{i}{c_{i}(0)}} + {\beta_{i}\theta} + \frac{\beta_{i}}{b_{i}}} < 0},$

then x_(i)(θ)=0; if

${{1 - {\left( {b_{i} - {\beta_{i}x_{i}^{+}}} \right){c_{i}^{\prime}\left( x_{i}^{+} \right)}} + {\beta_{i}{c_{i}\left( x_{i}^{+} \right)}} + {\beta_{i}\theta} + \frac{\beta_{i}}{b_{i} - {\beta_{i}x_{i}^{+}}}} > 0},$

then x_(i)(θ)=x_(i) ⁺). In addition, the optimal quality and pricevalues are given by

$\begin{matrix}{x_{i}^{*} = {x_{i}\left( \theta^{*} \right)}} & 27 \\{p_{i}^{*} = {{c_{i}\left( x_{i}^{*} \right)} + \frac{1}{b_{i} - {\beta_{i}x_{i\;}^{*}}} + {\theta^{*}.}}} & 28\end{matrix}$

It should be noted that, given additively separable cost functions andsymmetric b and β, the optimal quality is not identical across productsbut varies based on a_(i) values, which can be derived from equation(17). As a result, the optimal markup must also differ across productswith different a_(i) values due to equation (28). The followingcorollary provides a key insight into the implication of the interactionterm.

Corollary 7 Suppose Assumption 2 holds. In addition, assume that thecost function is additionally separable a; and x_(i), i.e.,c_(i)(x_(i))=c_(a)(a_(i))+c_(x)(x_(i)) where c_(a)(·) is anon-decreasing function, and that b_(i)=b and β_(i)=β for all i∈l. Then,x*_(i)≥x*_(i′) and *_(i)≥m*_(i′) if and only if a_(i)≥a_(i′) for any i,i′∈l.

Contrasting this with Corollary 4, the optimal quality levels andmarkups now differ by a_(i) values and products with a larger a_(i)value is matched with a higher quality as well as a higher markup. In apractical setting, this implies, for example, that the smart phonemanufacturer shall design its product line such that a premium model(which corresponds to a high a_(i) value) is matched with a premiumstorage size as well as a premium price—a commonly-adopted strategywhich can now be quantified and optimized with the model developed inthis paper.

From (21), it can also be observed that the properties of the optimalprices identified in Corollary 1 for the price optimization problemcontinue to hold for the joint optimization problem. When price andquality can be determined jointly, lower price sensitivity of a productallows the firm to charge a higher price for the product, andsubsequently to also set a higher quality value. Thus the relativemagnitude of x_(i) versus other products depends on both βi and bi, asshown in the following corollary.

Corollary 8 Suppose Assumption 2 holds. For any i, i′∈I and i≠i′,

let b _(i) =b _(i′), β_(i)=β_(i′) and c _(i)( )=c _(i′)(·), then x*_(i) >x* _(i′) if and only if p* _(i) >p* _(i′).  (i)

if β_(i)=β_(i′) and c _(i)(·)=c _(i′)(·), then x* _(i) >x* _(i′) if andonly if b _(i) <b _(i′.)  (ii)

In addition, we derive the following bounds for θ*.

Corollary 9 Under Assumption 2,

${\overset{\_}{\pi}}_{J} \leq \theta^{*} \leq {{\overset{\_}{\pi}}_{J} + {\frac{\sum_{i \in I}{e^{{ma}\; x{\{{{a_{i} + \frac{b_{i}}{\beta_{i}} - {\frac{b_{i}^{2}}{\beta_{i}}{c_{i}^{\prime}{(0)}}}},{a_{i} - 1 - {b_{i}{({{c_{i}{(0)}} + {\overset{\_}{\pi}}_{j}})}}}}\}}}/b_{i}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}.}}$

Theorem 7 identifies the fixed-point equation for the optimal solutionbut does not establish uniqueness of the solution or identify anefficient solution algorithm. Next, it can be shown that the fixed-pointsolution to (30) is unique and can be obtained with a bisection search.Define

${g(\theta)}:={\frac{\sum_{i \in I}{e^{{x_{i}{(\theta)}} + a_{i} - {{({b_{i} - {\beta_{i}{x_{i}{(\theta)}}}})}{c_{i}{({x_{i}{(\theta)}})}}} - 1 - {{({b_{i} - {\beta_{i}{x_{i}{(\theta)}}}})}\theta}}/\left( {b_{i} - {\beta_{i}{x_{i}(\theta)}}} \right)}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}.}$

Theorem 8. g(θ) monotonically decreases in θ and equation (30) has aunique fixed-point solution.

As a result of Theorem 8, the solution of equation (30) can be obtainedthrough an efficient bisection search algorithm.

Assumption 6. For any x*_(i)∈(0, x_(i) ⁺) that satisfies (24), the costfunction c_(i)(·) satisfies

$\begin{matrix}{{\left( {b_{i} - {\beta_{i}x_{i}^{*}}} \right){c_{i}^{''}\left( x_{i}^{*} \right)}} > {{2\beta_{i}{c_{i}^{\prime}\left( x_{i}^{*} \right)}} + {\left( \frac{\beta_{i}}{b_{i} - {\beta_{i}x_{i}^{*}}} \right)^{2}.}}} & 29\end{matrix}$

Assumption 6 implies for a given θ, the left side of Equation (24) canonly cross 1+β_(i)θ from below. Therefore, if a solution to Equation(24) exists, it must be unique. Under Assumption 6, the Jacobian matrixJ(x*) is a diagonal matrix with nonnegative diagonal elements (note that

$\left. {{\frac{\partial{h_{i}(x)}}{\partial x_{i}} = {{\left( {b_{i} - {\beta_{i}x_{i}^{*}}} \right){c_{i}^{''}\left( x_{i}^{*} \right)}} - {2\beta_{i}{c_{i}^{\prime}\left( x_{i}^{*} \right)}} - {\left( \frac{\beta_{i}}{b_{i} - {\beta_{i}x_{i}^{*}}} \right)^{2}\mspace{14mu} {and}}}}{\frac{\partial{h_{i}(x)}}{\partial x_{j}} = 0}} \right),$

which is positive semi-definite. This ensures global optimality.Theorem 9. If Assumptions 4 and 6 hold, then the optimal profit θ* tothe joint attribute and price optimization problem is the fixed-pointsolution to

$\begin{matrix}{\theta = {\frac{\sum_{i \in I}{e^{{x_{i}{(\theta)}} + a_{i} - {{({b_{i} - {\beta_{i}{x_{i}{(\theta)}}}})}{c_{i}{({x_{i}{(\theta)}})}}} - 1 - {{({b_{i} - {\beta_{i}{x_{i}{(\theta)}}}})}/\theta}}/\left( {b_{i} - {\beta_{i}{x_{i}(\theta)}}} \right)}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}} + {\frac{\sum_{j \in J}{{\overset{\_}{m}}_{j}e^{{\overset{\_}{u}}_{j}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}.}}} & 30\end{matrix}$

where x_(i)(θ) is the unique solution of Equation (24) for any given θif a solution to Equation (24) exists and x_(i)(θ)=x_(i) ⁺ otherwise. Inaddition, the optimal attribute and price values are given by

$\begin{matrix}{x_{i}^{*} = {x_{i}\left( \theta^{*} \right)}} & 31 \\{p_{i}^{*} = {{c_{i}\left( x_{i}^{*} \right)} + \frac{1}{b_{i} - {\beta_{i}x_{i}^{*}}} + {\theta^{*}.}}} & 32\end{matrix}$

It should be remarked that, given additively separable cost functionsand symmetric b and β, the optimal attribute is not identical acrossproducts but varies based on a_(i) values, which can be derived fromequation (24). As a result, the optimal markup must also differ acrossproducts with different a_(i) values due to equation (32). The followingcorollary provides a key insight into the implication of the interactionterm.

Corollary 10. Suppose Assumptions 4 and 6 hold. In addition, assume thatthe cost function is additively separable in a_(i) and x_(i), i.e.,c_(i)(x_(i))=c_(a)(a_(i))+c_(x)(x_(i)) where c_(a)(·) is anon-decreasing function, and that b_(i)=b and β_(i)=β for all i∈l. Thenx*_(i)≥x*_(i′) and m*_(i)≥m*_(i′) if and only if a_(i)≥a_(i′) for any i,i′∈l.

Contrasting this with Corollary 6, the optimal attributes and markupsnow differ by a_(i) values and products with a larger a_(i) value ismatched with a higher attribute as well as a higher markup. In apractical setting, this implies, for example, that the smart phonemanufacturer shall design its product line such that a premium model(which corresponds to a high a_(i) value) is matched with a premiumstorage size as well as a premium price—a commonly-adopted strategywhich can now be quantified and optimized with the model developed inthis paper.

From Equation (32), it is also observed that the properties of theoptimal prices identified in Corollary 1 for the price optimizationproblem continue to hold for the joint optimization problem. However,the properties of the optimal attribute values given in Corollary 4 arenot always retained in the joint optimization problem. In particular,while property (i) in Corollary 4 largely stays true, properties (ii)and (iii) both break down under joint optimization.

Corollary 11. Suppose Assumptions 4 and 6 hold. For any i, i′∈l andi≠i′,

let b _(i) =b _(i′), β_(i) =βi′ and c′ _(i)(·)=c′ _(i′)(·), then x*_(i) >x* _(i′) if and only if p* _(i) >p* _(i′).  (i)

if β_(i)=β_(i′) and c _(i)(·)=c _(i′)(·), then x* _(i) >x* _(i′) if andonly if b _(i) <b _(i′).  (ii)

When price and attribute can be determined jointly, lower pricesensitivity of a product allows the firm to charge a higher price forthe product, and subsequently to also set a higher attribute value. Thusthe relative magnitude of x_(i) versus other products becomes dependenton both β_(i) and b_(i), which explains why properties (ii) and (iii) inCorollary 4 do not carry through to joint optimization.

From equation (24), the optimal solution satisfies

$\begin{matrix}{{\theta^{*} + {c_{i}\left( x_{i}^{*} \right)}} = {{{\frac{b_{i} - {\beta_{i}x_{i}^{*}}}{\beta_{i}}{c_{i}^{\prime}\left( x_{i}^{*} \right)}} - {\left( {\frac{1}{\beta_{i}} + \frac{1}{b_{i} - {\beta_{i}x_{i}^{*}}}} \right)\mspace{14mu} {for}\mspace{14mu} {all}\mspace{14mu} i}} \in I}} & 33\end{matrix}$

Substitute the above into Equation (26),

${\left( {1 + {\sum\limits_{j \in J}e^{{\overset{\_}{u}}_{j}}}} \right)\left( {\theta^{*} - \frac{\sum_{j \in J}{{\overset{\_}{m}}_{j}e^{{\overset{\_}{u}}_{j}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}} \right)} = {{\sum\limits_{i \in I}{e^{x_{i}^{*} + a_{i} - 1 - {{({b_{i} - {\beta_{i}x_{i}^{*}}})}{\lbrack{{\frac{b_{i} - {\beta_{i}x_{i}^{*}}}{\beta_{i}}{c_{i}^{\prime}{(x_{i}^{*})}}} - {({\frac{1}{\beta_{i}} + \frac{1}{b_{i} - {\beta_{i}x_{i}^{*}}}})}}\rbrack}}}/\left( {b_{i} - {\beta_{i}x_{i}^{*}}} \right)}} = {{\sum\limits_{i \in I}{e^{a_{i} + \frac{b_{i}}{\beta_{i}} - {\frac{({b_{i} - {\beta_{i}x_{i}^{*}}})}{\beta_{i}}{c_{i}^{\prime}{(x_{i}^{*})}}}}/\left( {b_{i} - {\beta_{i}x_{i}^{*}}} \right)}} = {\sum\limits_{i \in I}{{\exp \left( {a_{i} + \frac{b_{i}}{\beta_{i}}} \right)}{{\exp \left( {{- {\log \left( {b_{i} - {\beta_{i}x_{i}^{*}}} \right)}} - {\frac{\left( {b_{i} - {\beta_{i}x_{i}^{*}}} \right)^{2}}{\beta_{i}}{c_{i}^{\prime}\left( x_{i}^{*} \right)}}} \right)}.}}}}}$

It can be shown that under Assumption 6, the term

${- {\log \left( {b_{i} - {\beta_{i}x_{i}^{*}}} \right)}} - {\frac{\left( {b_{i} - {\beta_{i}x_{i}^{*}}} \right)^{2}}{\beta_{i}}{c_{i}^{\prime}\left( x_{i}^{*} \right)}}$

is strictly decreasing in x*_(i). Therefore,

${{\left( {1 + {\sum\limits_{j \in J}e^{{\overset{\_}{u}}_{j}}}} \right)\left( {\theta^{*} - \frac{\sum_{j \in J}{{\overset{\_}{m}}_{j}e^{{\overset{\_}{u}}_{j}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}} \right)} \leq {\sum\limits_{i \in I}{{\exp \left( {a_{i} + \frac{b_{i}}{\beta_{i}}} \right)}{\exp \left( {{- {\log \left( b_{i} \right)}} - {\frac{b_{i}^{2}}{\beta_{i}}{c_{i}^{\prime}(0)}}} \right)}}}} = {\sum\limits_{i \in I}{\frac{1}{b_{i}\;}{{\exp \left( {a_{i} + \frac{b_{i}}{\beta_{i}} - {\frac{b_{i}^{2}}{\beta_{i}}{c_{i}^{\prime}(0)}}} \right)}.}}}$

Equivalently,

$\theta^{*} \leq {\frac{\sum_{i \in I}{\frac{1}{b_{i}}{\exp \left( {a_{i} + \frac{b_{i}}{\beta_{i}} - {\frac{b_{i}^{2}}{\beta_{i}}{c_{i}^{\prime}(0)}}} \right)}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}} + \frac{\sum_{j \in J}{{\overset{\_}{m}}_{j}e^{{\overset{\_}{u}}_{j}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}}$

Therefore, the following bounds are derived for θ*.

Corollary 12.

${\overset{\_}{\pi}}_{j} \leq \theta^{*} \leq {{\overset{\_}{\pi}}_{J} + {\frac{\sum_{i \in I}{e^{a_{i} + {m\; i\; n{\{{{x_{i}^{+} - 1},{\frac{b_{i}}{\beta_{i}} - {\frac{b_{i}^{2}}{\beta_{i}}{c_{i}^{\prime}{(0)}}}}}\}}}}/b_{i}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}.}}$

It is possible that for certain θ values, equation (24) will not have asolution. This could occur in one of the following cases: θ value isunreasonably large, or

${\left( {b_{i} - {\beta_{i}x_{i}}} \right){c_{i}^{\prime}\left( x_{i} \right)}} - {\beta_{i}\left( {{c_{i}\left( x_{i} \right)} + \frac{1}{b_{i} - {\beta_{i}x_{i}}}} \right)}$

is too small, or both (for example, if

${\left( {b_{i} - {\beta_{i}x_{i}}} \right){c_{i}^{\prime}\left( x_{i} \right)}} - {\beta_{i}\left( {{c_{i}\left( x_{i} \right)} + \frac{1}{b_{i} - {\beta_{i}x_{i}}}} \right)}$

for all x_(i)∈(0, x_(i) ⁺], then (24) is infeasible). In this case, itis easy to see from (22) that

$\frac{\partial{\theta (x)}}{{\partial x_{i}}\;} > 0$

and the optimal solution is to set x_(i) as large as possible, i.e., atx_(i) ⁺. Therefore, in numerical implementation, if infeasibility occursat a given value of θ, then let x_(i)(θ)=x_(i) ⁺.Theorem 10-. The solution to equation (26) is unique.

The proof of Theorem 10 relies on showing that the right side of (26)strictly decreases in θ. As a result, the solution can be obtainedthrough an efficient bisection search algorithm.

Algorithm 3. (Attribute and Price Optimization)${1.\mspace{14mu} {Let}\mspace{14mu} \theta^{-}} = {{\frac{\sum_{j \in J}{{\overset{\_}{m}}_{j}e^{{\overset{\_}{u}}_{j}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}\mspace{14mu} {and}\mspace{14mu} \theta^{+}} = {\frac{\sum_{i \in I}e^{x_{i}^{+} + a_{i} - {1/b_{i}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}} + \frac{\sum_{j \in J}{{\overset{\_}{m}}_{j}e^{{\overset{\_}{u}}_{j}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}}}$2. Let θ = (θ⁻ + θ⁺)/2 and solve (24) for x_(i)(θ), i ∈ I followingSteps (a)-(e):  (a) Let y⁻ = 0 and y⁺ = x_(i) ⁺  (b) Let y = (y⁻ +y⁺)/2.   ${(c)\mspace{14mu} {Compute}\mspace{14mu} g} = {{\left( {b_{i} - {\beta_{i}x_{i}}} \right){c_{i}^{\prime}\left( x_{i} \right)}} - {{\beta_{i}\left( {{c_{i}\left( x_{i} \right)} + \frac{1}{b_{i} - {\beta_{i}x_{i}}}} \right)}.}}$ (d) if g > 1 + β_(i)θ, let y⁻ = y; if g < 1 + β_(i)θ, let y⁺ = y.  (e)Repeat Steps (a)-(e) until g = θ. Then x_(i)(θ) = y. 3. If, for thegiven θ value, Steps (a)-(e) do not converge, then let θ⁺ = θ and repeatStep 2. If no solution is found for θ at its lowest value, then setx_(i) ^(*) = x_(i) ⁺.${{4.\mspace{14mu} {Let}\mspace{14mu} {c_{i}\left( x_{i} \right)}} = {c_{i}\left( {x_{i}(\theta)} \right)}},{{i \in {I\mspace{14mu} {and}\mspace{14mu} {compute}\mspace{14mu} f}} = {\frac{\sum_{i \in I}e^{x_{i} + a_{i} - {{({b_{i} - {\beta_{i}x_{i}}})}{c_{i}{(x_{i})}}} - 1 - {{({b_{i} - {\beta_{i}x_{i}}})}^{\theta}/{({b_{i} - {\beta_{i}x_{i}}})}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}} + {\frac{\sum_{j \in J}{{\overset{\_}{m}}_{j}e^{{\overset{\_}{u}}_{j}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}.}}}$5. if f > θ, let θ⁻ = θ; if f < θ, let θ⁺ = θ. 6. Repeat Steps 2-4 untilf = θ.

Lastly, it should be remarked that the condition in Assumption 6 relieson the specification of the cost function c_(i)(x_(i)). Given the costfunction form, the condition may be further simplified or, in somesettings, used to generate simpler sufficient conditions.

Algorithm 4. (Quality and Price Optimization)${1.\mspace{14mu} {Let}\mspace{14mu} \theta^{-}} = {{{\overset{\_}{\pi}}_{J}\mspace{14mu} {and}\mspace{14mu} \theta^{+}} = {{\overset{\_}{\pi}}_{J} + {\frac{\sum_{i \in I}{e^{\max {\{{{a_{i} + \frac{b_{i}}{\beta_{i}} - {\frac{b_{i}^{2}}{\beta_{i}}{c_{i}^{\prime}{(0)}}}},{a_{i} - 1 - {b_{i}{({{c_{i}{(0)}} + {\overset{\_}{\pi}}_{J}})}}}}\}}}\text{/}b_{i}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}.}}}$2. Let θ = (θ⁻ + θ⁺)/2 and solve (17) for x_(i)(θ), i ∈ I followingSteps (a)-(e):  (a) Let y⁻ = 0 and y⁺ = x_(i) ⁺.  (b) Let y = (y⁻ +y⁺)/2.   ${(c)\mspace{14mu} {Compute}\mspace{14mu} z} = {{\left( {b_{i} - {\beta_{i}x_{i}}} \right){c_{i}^{\prime}\left( x_{i} \right)}} - {{\beta_{i}\left( {{c_{i}\left( x_{i} \right)} + \frac{1}{b_{i} - {\beta_{i}x_{i}}}} \right)}.}}$ (d) if z > 1 + β_(i)θ, let y⁻ = y; if z < 1 + β_(i)θ, let y⁺ = y.  (e)Repeat Steps (a)-(e) until z = θ or y⁺ = y⁻. Then x_(i)(θ) = y. 3. If,for the given θ value, Steps (a)-(e) do not converge, then let θ⁺ = θand repeat Step 2. If no solution is found for θ at its lowest value,then set x_(i) ^(*) = x_(i) ⁺.${{4.\mspace{14mu} {Let}\mspace{14mu} {c_{i}\left( x_{i} \right)}} = {c_{i}\left( {x_{i}(\theta)} \right)}},{{i \in {I\mspace{14mu} {and}\mspace{14mu} {compute}\mspace{14mu} f}} = {\frac{\sum_{i \in I}e^{x_{i} + a_{i} - {{({b_{i} - {\beta_{i}x_{i}}})}{c_{i}{(x_{i})}}} - 1 - {{({b_{i} - {\beta_{i}x_{i}}})}^{\theta}/{({b_{i} - {\beta_{i}x_{i}}})}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}} + {{\overset{\_}{\pi}}_{J}.}}}$5. if f > θ, let θ⁻ = θ; if f < θ, let θ⁺ = θ. 6. Repeat Steps 2-4 untilf = θ.

If, for a given cost function, Assumption 5 is not satisfied, thenAlgorithm 4 may not be applied. Instead, a special characteristic of theoptimality condition in (17) may be noted. That is, given the value ofθ, the left side of the equation does not depend on x_(j), j≠i. Thisimplies that, given θ, a single-dimension search can be used to find allstationary x_(i) values for all i∈I. For any given θ, let x_(i) ^(k)(θ),k=1, . . . , K where K>1 be the multiple paths of solutions to equation(17). For any given path (defined by some selection rule when picking asolution to (17) from potentially multiple possibilities), we can showthat g(θ) is decreasing in θ by applying Theorem 8 for thisgeneralization. In other words, we have multiple g functions, i.e.,g¹(θ), . . . , g^(K)(θ) which are decreasing in θ. The function thatyields the largest fixed-point solution to (30) yields the globalmaximum. Denote this function with g*(θ) and the global maximum with θ*.

Since each g^(k)(θ) function decreases monotonically in θ, it is easy tosee that the global maximum must satisfy g^(k)(θ*)≥g^(k)(θ*) for allk=1, . . . , K. Consequently, to locate the global maximum, it sufficesto locate the fixed-point solution of θ=g_(max)(θ)+π _(J) whereg_(max)(θ):=max_(k) g^(k)(θ). Since gmax(θ) must also be decreasing inθ, we can apply bisection search to obtain the optimal value of θ.Therefore, we propose the following algorithm for obtaining the optimalsolution when Assumption 5 does not hold or cannot be verified.

Algorithm 5. (Quality and Price Optimization without Assumption 5)${{{{1.\mspace{14mu} {Let}\mspace{14mu} \theta^{-}} = {{{\overset{\_}{\pi}}_{J}\mspace{14mu} {and}\mspace{14mu} \theta^{+}} =}}\quad}\quad}{\quad{{\overset{\_}{\pi}}_{J} + {\frac{\sum_{i \in I}e^{x_{i}^{+} + a_{i} - {{({b_{i} - {\beta_{i}x_{i}^{+}}})}{c_{i}{(0)}}} - 1 - {{({b_{i} - {\beta_{i}x_{i}^{+}}})}{\theta^{-}/{({b_{i} - {\beta_{i}x_{i}^{+}}})}}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}}.}}}$2. Let θ = (θ⁻ + θ⁺)/2.  (a) Search in the range of [0,x_(i) ⁺] for allvalues of x_(i) that satisfy (17) and place them in set X_(i). Inaddition, place 0 and x_(i) ⁺ in set X_(i), if they are not alreadyincluded.   ${{{{{(b)\mspace{14mu} {For}\mspace{14mu} {each}\mspace{14mu} x} = {{\left( x_{i} \right)_{i \in I}\mspace{14mu} {where}\mspace{14mu} x_{i}} \in _{i}}}\;, {compute}}\mspace{20mu}\quad}{f(x)}} = {\frac{\sum_{i \in I}e^{x_{i} + a_{i} - {{({b_{i} - {\beta_{i}x_{i}}})}{c_{i}{(x_{i})}}} - 1 - {{({b_{i} - {\beta_{i}x_{i}}})}{\theta/{({b_{i} - {\beta_{i}x_{i}}})}}}}}{1 + {\sum_{j \in J}e^{{\overset{\_}{u}}_{j}}}} + {{\overset{\_}{\pi}}_{J}.}}$ (c) Let f_(max) = max_(x) _(i) _(∈X) _(i) ^(,) _(i∈I) f(x). 3. Iff_(max) > θ, then let θ⁻ = θ; if f_(max) < θ, then let θ⁺ = θ. 4. RepeatSteps 2-3 until f_(max) = θ.

It should be noted that the upper bound from Corollary 9 holds true onlyunder Assumption 5. Therefore in Algorithm 5 an upper bound can beconstructed based solely on x_(i) ∈[0, x_(i) ⁺].

EXAMPLES

Consider a manufacturer with the product cost function c(a,x)=0.1a+0.1e^(1.5x). The manufacturer has an existing set of productswith {a_(j)}_(j∈J)=[0, 5, 10], {x _(j)}_(j∈J)=[0.5, 0.8, 1], {p_(j)}_(j∈J)=[2, 3, 4]. It introduces three new products with{a_(i)}_(i∈l)=[, 5, 10] and jointly optimizes prices and attributes ofthe new products in the expanded product line. The values of b and β arethe same across products and are given in Table 6. Algorithm 3 isapplied to optimize both prices and attributes (i.e., p_(i), x_(i), i∈l)of the new products and the results are recorded in Table 6.

TABLE 6 Jointly Optimization of Attributes and Prices. OptimalAttributes Optimal Prices Instance b β x₄* x₅* x₆* p₄* p₅* p₆* profit 13.0 0.35 1.08 1.15 1.22 2.85 3.41 3.97 1.96 2 3.0 0.40 1.17 1.25 1.323.03 3.60 4.17 2.05 3 3.0 0.45 1.27 1.35 1.42 3.22 3.81 4.40 2.14 4 3.50.35 0.84 0.92 0.99 2.01 2.56 3.11 1.35 5 3.5 0.40 0.91 0.99 1.07 2.112.66 3.22 1.40 6 3.5 0.45 0.98 1.07 1.15 2.22 2.78 3.35 1.45 7 4.0 0.350.66 0.75 0.82 1.52 2.05 2.59 0.98 8 4.0 0.40 0.71 0.80 0.89 1.57 2.112.66 1.01 9 4.0 0.45 0.77 0.86 0.95 1.62 2.18 2.73 1.03

Note that products 4-6 are differentiated only by their a, value priorto the optimization. The optimized prices and attributes follow asequence matching that of a/s, as Corollary 10 implies.

Recall that, under attribute optimization, the optimal attributesincrease with price sensitivity b. This trend is reversed with jointoptimization as illustrated in Table 6 and FIGS. 2A and 2B: as customersbecome more price sensitive (equivalently, less willing to pay), themanufacturer lowers prices, which, due to the interaction of attributeand price, leads to lower marginal utility of attribute (equation (2));this, consequently, drives the manufacturer to reduce attribute.

Now consider the effect of β. Larger β implies higher marginal utilityof x, creating incentive to increase attribute; the increased attributein turn, reduces customers' marginal disutility of price and thus drivesup optimal prices, which in turn further increases the marginal utilityof x. Due to such a reinforcement effect on the marginal utility,increasing β leads to both higher optimal attributes and higher optimalprices. This stands in contrast to the observation in FIG. 1B forattribute optimization where fixed prices lead to diminishing returnsfrom attribute at higher β values. This discussion reveals a centralrole that the interaction effect plays in the joint optimization ofattributes and prices, i.e., it causes the optimal prices and attributesto move in tandem, and as the interaction intensifies, both move upward.

Joint optimization dominates price or attribute optimization alone andleads to higher profit. We illustrate such improvement with numericalexperiments that consider multiple b and β parameter combinations. Foreach parameter combination, we generate 100 random problem instances bydrawing the attribute values of the new products from a uniformdistribution on [0, 2] and perform price optimization for each instance.We then compute the percentage profit improvement of joint optimizationover price optimization and average over these 100 instances; we alsopresent the average optimal profit underprice optimization and theoptimal profit under joint optimization (although we note that thepercentage improvement is not based on the average optimal profit). Seeresults in Table 7. Similarly, for each b, β parameter combination, wegenerate 100 problem instances by randomly selecting the price of eachnew product within ±15% range of the price for a comparable existingproduct (i.e., those with the same a value) and perform attributeoptimization. Profit improvement with joint optimization is thencomputed and averaged over these 100 instances; see results in Table 8.As is shown, profit improvement with joint optimization can besubstantial.

Extension: Multi-Dimensional Attribute

Thus far, optimization of a single-dimension attribute has beenillustrated and discussed. One can easily envision a practical scenarioin which a firm optimizes attribute values across multiple dimensions.For instance, the smartphone manufacturer may differentiate model M2products in both storage size and screen size or some other potentialfeatures. In this section, the model is enhanced to addressmulti-dimensional attribute.

TABLE 7 Profit Improvement (Joint vs. Price Optimization). AverageProfit for Profit for Price Joint Average Combination b β OptimizationOptimization Improvement 1 3.5 0.35 1.23 1.35 11.1% 2 3.5 0.40 1.28 1.409.8% 3 3.5 0.45 1.34 1.45 8.9% 4 4.5 0.35 0.60 0.74 35.8% 5 4.5 0.400.62 0.76 31.8% 6 4.5 0.45 0.65 0.77 28.2% 7 5.5 0.35 0.31 0.43 91.0% 85.5 0.40 0.32 0.44 81.0% 9 5.5 0.45 0.33 0.45 73.2%

TABLE 8 Profit Improvement (Joint vs. Attribute Optimization). AverageProfit for Profit for Attribute Joint Average Combination b βOptimization Optimization Improvement 1 3.5 0.35 1.07 1.35 29% 2 3.50.40 1.20 1.40 18% 3 3.5 0.45 1.32 1.45 11% 4 4.0 0.35 0.36 0.98 220% 54.0 0.40 0.45 1.01 158% 6 4.0 0.45 0.54 1.03 112% 7 4.5 0.35 0.08 0.741188% 8 4.5 0.40 0.11 0.76 883% 9 4.5 0.45 0.14 0.77 648%

The utility of product i by a randomly selected customer is

u _(i) =x _(i)(y _(i))−b _(i) p _(i)+β_(i) p _(i) x(y _(i))+a _(i)+ϵ_(i)

where x_(i)(y_(i))=y_(i1)+ . . . +y_(im)=y_(i) ^(T)1 (1 is anm-dimensional vector of 1's) is the aggregate utility of non-priceattributes y_(i1), . . . , y_(im). Let c_(i)(y_(i)) denote the unit costof product i as a function of the vector of non-price attributesy_(i)=(y_(i1), y_(i2), . . . y_(im)), and let y=(y_(i), . . . , y_(m)).Thus, the purchase probabilities are

$\begin{matrix}{{q_{i}\left( {y,p} \right)} = {\frac{e^{{x_{i}{(y_{i})}} + a_{i} - {b_{i}p_{i}} + {\beta_{i}{x_{i}{(y_{i})}}p_{i}}}}{1 + {\sum\limits_{j = 1}^{n}e^{{x_{j}{(y_{j})}} + a_{j} - {b_{j}p_{j}} + {\beta_{j}{x_{j}{(y_{j})}}p_{j}}}}}\mspace{14mu} {and}}} & 34 \\{{q_{0}\left( {y,p} \right)} = {\frac{1}{1 + {\sum\limits_{j = 1}^{n}e^{{x_{j}{(y_{j})}} + a_{j} - {b_{j}p_{j}} + {\beta_{j}{x_{j}{(y_{j})}}p_{j}}}}}.}} & 35\end{matrix}$

The profit function is

$\pi = {{\sum\limits_{i \in I}{\left( {{\overset{\_}{p}}_{i} - {c_{i}\left( y_{i} \right)}} \right){q_{i}\left( {y,p} \right)}}} + {\sum\limits_{j \in J}{{\overset{\_}{m}}_{j}{{q_{j}\left( {y,p} \right)}.}}}}$

Define

$\begin{matrix}{{{\hat{q}}_{i}(x)} = \frac{e^{x_{i} + a_{i} - {b_{i}p_{i}} + {\beta_{i}x_{i}p_{i}}}}{1 + {\sum\limits_{j = 1}^{n}e^{x_{j} + a_{j} - {b_{j}p_{j}} + {\beta_{j}x_{j}p_{j}}}}}} & 36 \\{{{\hat{q}}_{0}(x)} = {\frac{1}{1 + {\sum\limits_{j = 1}^{n}e^{x_{j} + a_{j} - {b_{j}p_{j}} + {\beta_{j}x_{j}p_{j}}}}}.}} & 37\end{matrix}$

where x=(x_(i))_(i∈l). In addition, define

${{\hat{c}}_{i}\left( x_{i} \right)} = {\min\limits_{y_{i}}{\left\{ {\left. {c_{i}\left( y_{i} \right)} \middle| {y_{i}^{\prime}1} \right. = x_{i}} \right\}.}}$

The attribute optimization problem is rewritten as

${\max\limits_{x}{\pi (x)}} = {{\sum\limits_{i \in I}{\left( {p_{i} - {{\hat{c}}_{i}\left( x_{i} \right)}} \right){{\hat{q}}_{i}(x)}}} + {\sum\limits_{j \in J}{{\overset{\_}{m}}_{j}{{\hat{q}}_{j}(x)}}}}$

and the joint optimization problem as

${\max\limits_{x,p}{\pi \left( {x,p} \right)}} = {{\sum\limits_{i \in I}{\left( {p_{i} - {{\hat{c}}_{i}\left( x_{i} \right)}} \right){{\hat{q}}_{i}\left( {x,p} \right)}}} + {\sum\limits_{j \in J}{{\overset{\_}{m}}_{j}{{{\hat{q}}_{j}\left( {x,p} \right)}.}}}}$

Assumption 7. c_(i)(y_(i)) is convex on R^(n) and increasing in eachdimension, i.e.,

$\frac{\partial{c_{i}\left( y_{i} \right)}}{\partial y_{ik}} > 0$

for i∈l and k=1, . . . , m.Lemma 5. Suppose Assumption 7 holds. Then is increasing and convex.

The condition in Assumption 7 is satisfied by, for example, additivelyseparable cost functions c_(i)(y_(i))=c_(i1)(y₁)+c_(i2)(y₂)+ . . .+c_(im)(y_(m)) where c_(ik)(y_(k)), k=1, . . . , m is increasing andconvex, as well as non-separable cost functions that are increasing andconvex.

From Lemma 5, all results for the single-dimension attributeoptimization hold for multi-dimensional attributes under Assumption 7.Hence Algorithm 2 can be adopted for multi-dimension attributeoptimization by replacing the cost function with ĉ_(i)(·). To computeĉ′_(i)(x_(i)) and ĉ″_(i)(x_(i)), we follow a two-step procedure: (1)Given x_(i), solve the optimization problem min_(y) _(i){c_(i)(y_(i))|y′_(i)1=x_(i)} to obtain y*_(i). (2) Letĉ′_(i)(x_(i))=λ_(i) where λ_(i) is the Lagrangian multiplier for theconstraint y′_(i)1−x_(i) and compute

${{\hat{c}}_{i}^{''}\left( x_{i} \right)} = {\sum\limits_{k = 1}^{m}{\sum\limits_{j = 1}^{m}{\frac{\partial^{2}{c_{i}\left( y_{i}^{*} \right)}}{{\partial y_{ik}}{\partial y_{ij}}}.}}}$

For joint price-attribute optimization, the condition in Assumption 6 isrequired for the cost function ĉ(·):

$\begin{matrix}{{\left( {b_{i} - {\beta_{i}x_{i}^{*}}} \right){{\hat{c}}_{i}^{''}\left( x_{i}^{*} \right)}} > {{2\beta_{i}{{\hat{c}}_{i}^{\prime}\left( x_{i}^{*} \right)}} + {\left( \frac{\beta_{i}}{b_{i} - {\beta_{i}x_{i}^{*}}} \right)^{2}.}}} & 38\end{matrix}$

which translates to

$\begin{matrix}{{\left( {b_{i} - {\beta_{i}{\sum\limits_{k = 1}^{m}y_{ik}^{*}}}} \right){\sum\limits_{k = 1}^{m}{\sum\limits_{j = 1}^{m}\frac{\partial^{2}{c_{i}\left( y_{i}^{*} \right)}}{{\partial y_{ik}}{\partial y_{ij}}}}}} > {{2\beta_{i}\lambda_{i}} + {\left( \frac{\beta_{i}}{b_{i} - {\beta_{i}{\overset{m}{\sum\limits_{k = 1}}y_{ik}^{*}}}} \right)^{2}.}}} & 39\end{matrix}$

in terms of each individual attribute dimension. When condition (38)holds, all theoretical results and algorithm for single attribute jointprice-attribute optimization hold in the case of multi-dimensionalattribute. In practice, one of the main tasks of product design andengineering is to accurately estimate product cost based on materialcost and production technology. With multi-dimensional attribute, thefunction ĉ_(i)(x_(i)) can be viewed as an efficient frontier of productcosts and thus can be estimated accordingly. Consequently, verificationof (39) reduces to verifying condition (38) directly.

Effect of Existing Products on Optimal Decision

The impact of existing products on the optimal quality and pricedecisions of new products is explored, as well as how the impact ismodified by the price-quality interaction.

Recall that I is the set of new products and J is the set of existingproducts. Let π*_(I∪J) denote the total profit of the product line giventhe set of existing product J and that the price and quality of the newproducts in I are optimally determined. Let π*_(I) denote the optimalprofit with no existing product. Similarly, let (p*_(i|I∪P),x*_(i|I∪J))_(i∈I) and (p*_(i|I′), x*_(i|I))_(i∈I) denote the optimaldecisions with and without the existing products.

The following corollaries show the effect of existing products onquality/price decisions with and without price-quality interaction.Corollary 13 follows directly from Theorem 9. Corollary 14 followsdirectly from Theorem 7 and the fact that the left-hand side of equation(17) is increasing in x under Assumption 5.

Corollary 13. Suppose β_(i)=0 for all i∈I. (i) The optimal quality ofthe new product x*_(i), i⊂I is independent of price and quality of anyexisting product. (ii) If π*_(I∪J)>π*_(I), then the presence of existingproducts J causes the prices of new products to increase, i.e., forp_(i|l∪J) ⁺>p_(i|I) ⁺ for i∈I: otherwise, the opposite holds true.

Corollary 14. Suppose βi>0 for all i∈I and Assumption 5 holds. Ifπ*_(I∪J)>πI_(I), then the presence of existing products J causes boththe quality and prices of new products to increase, i.e.,x*_(i|I∪J)>x*_(x|l) and p*_(i|I∪J)>p*_(∈I) for all i∈I; otherwise, theopposite holds true.

The effect of existing products on the optimal price/quality decisionsfor new products is determined by the relationship between π*_(I∪J) andπ*_(J) . From a practical perspective, the existing products J thatremain in the offer set at the time new products are introduced are suchthat π*_(IÅJ)>π*_(I), i.e., if π*_(I∪J)<π*_(I), then one or more of theexisting products should be dropped from the product line.

Without interaction, assuming that the firm has made the right decisionfor including products j∈J, the presence of existing products drives upthe optimal prices of the new products, but does not affect the optimalquality of the new products. The independence of optimal quality fromexisting products is a consequence of zero price-quality interaction andis arguably unrealistic in most industry contexts. With interaction, thepresence of existing products (assuming that inclusion is a gooddecision) ought to drive the firm to offer new products positionedhigher in both quality and price. This resonates with practicalobservations. For example, the latest iPhone model X has been introducedin the presence of existing iPhone7 and iPhone8 products, priced athefty $999, $1149, and $1349 for 64 GB, 256 GB, and 512 GB storage sizerespectively and a slew of other fancy high-end features (AppleCorporation Website, 2018a). Had iPhone7 and iPhone8 not been included,Apple would probably not have aimed its new products at such extremehigh-end target position.

Conclusion and Discussion

Constantly evolving product lines create challenges for product design.In this paper, we address this complex problem by formulating thepricing and attribute decision problem using a MNL model withattribute-price interaction. We consider three practical variations ofthe problem: (i) optimize prices of the new products in the presence ofexisting products in the product line and pre-determined productattributes of the new products, (ii) optimize attributes of the newproducts in the presence of existing products and pre-determined pricesof the new products, and (iii) optimize both prices and attributes ofthe new products in the presence of existing products. The profitfunction and the optimal solution are characterized, in particular, howthe optimal attributes and/or optimal prices vary across products andwith the parameters. Our analysis yields efficient solution algorithmsfor each problem variation.

An important message that this disclosure brings forth is that the lackof realism in the linear utility of the MNL model and the resultingequal markup and equal attribute properties can be addressed with aninteraction term. This interaction term is a simple but powerfulextension that is central to understanding the attribute and pricedecision in product line design. With the interaction effect, theoptimal attribute and markup vary across products even under identicalprice sensitivity and cost function. Illustrative examples add furtherinsights on how the optimal solution is affected by coefficients of theutility model and how the joint optimization improves the firm's profitbeyond what is accomplished by price or attribute optimization alone.

In practice, a mixed optimization in which price optimization andattribute optimization may be performed on different sets of products islikely. For example, the manufacturer may decide to optimize prices ofboth new and old products but only optimize attributes of the newproducts. This scenario is not modeled in the current three variations;however, our model can be extended to consider such a scenario. To seethis, note that given attributes, price optimization yields a uniqueprice solution. That is, given x_(i) there exists θ(x) that solves

$\begin{matrix}{\theta = {{\sum\limits_{i \in I}{e^{x_{i} + a_{i} - {{({b_{i} - {\beta_{i}x_{i}}})}{({\theta + {c_{i}{(x_{i})}}})}} - 1}/\left( {b_{i} - {\beta_{i}x_{i}}} \right)}} + {\sum\limits_{j \in J}{e^{{\overset{\_}{x}}_{j} + a_{j} - {{({b_{j} - {\beta_{j}{\overset{\_}{x}}_{j}}})}{({\theta + c_{j}})}} - 1}/{\left( {b_{j} - {\beta_{j}{\overset{\_}{x}}_{j}}} \right).}}}}} & 40\end{matrix}$

Thus Theorem 5 holds by replacing equation (19) with (36). We cansubsequently show that under Assumptions 4 and 6, the optimal solutionof the joint price-attribute decision is unique and given by the fixedpoint equation

$\theta = {{\sum\limits_{i \in I}{e^{{x_{i}{(\theta)}} + a_{i} - {{({b_{i} - {\beta_{i}{x_{i}{(\theta)}}}})}{({\theta + {c_{i}{({x_{i}{(\theta)}})}}})}} - 1}/\left( {b_{i} - {\beta_{i}{x_{i}(\theta)}}} \right)}} + {\sum\limits_{j \in J}{e^{{\overset{\_}{x}}_{j} + a_{j} - {{({b_{j} - {\beta_{j}{\overset{\_}{x}}_{j}}})}{({\theta + c_{j}})}} - 1}/{\left( {b_{j} - {\beta_{j}{\overset{\_}{x}}_{j}}} \right).}}}}$

Hence Algorithm 3 holds for the mixed optimization with the aboveadjustment. Lastly, we remark that, although we model the fixedattribute a_(i) without an interaction with price, it should be clearthat explicitly adding the interaction of a_(i) with p_(i) results inthe same problem formulation. In sum, the model and methods in thisdisclosure apply broadly to practical decision scenarios.

Computing Device

FIG. 3 illustrates an example of a suitable computing device 200 whichmay be used to implement various aspects of a pricing model and one ormore corresponding solution algorithms to generate optimized pricing, asdescribed herein. More particularly, in some embodiments, aspects of thedescribed optimizing pricing model may be translated to software ormachine-level code, which may be installed to and/or executed by thecomputing device 200 such that the computing device 200 is configured togenerate optimized pricing according to the methods and functionsdescribed herein. It is contemplated that the computing device 200 mayinclude any number of devices, such as personal computers, servercomputers, hand-held or laptop devices, tablet devices, multiprocessorsystems, microprocessor-based systems, set top boxes, programmableconsumer electronic devices, network PCs, minicomputers, mainframecomputers, digital signal processors, state machines, logic circuitries,distributed computing environments, and the like.

The computing device 200 may include various hardware components, suchas a processor 202, a main memory 204 (e.g., a system memory), and asystem bus 201 that couples various components of the computing device200 to the processor 202. The system bus 201 may be any of several typesof bus structures including a memory bus or memory controller, aperipheral bus, and a local bus using any of a variety of busarchitectures. For example, such architectures may include IndustryStandard Architecture (ISA) bus, Micro Channel Architecture (MCA) bus,Enhanced ISA (EISA) bus, Video Electronics Standards Association (VESA)local bus, and Peripheral Component Interconnect (PCI) bus also known asMezzanine bus.

The computing device 200 may further include a variety of memory devicesand computer-readable media 207 that includes removable/non-removablemedia and volatile/nonvolatile media and/or tangible media, but excludestransitory propagated signals. Computer-readable media 207 may alsoinclude computer storage media and communication media. Computer storagemedia includes removable/non-removable media and volatile/nonvolatilemedia implemented in any method or technology for storage ofinformation, such as computer-readable instructions, data structures,program modules or other data, such as RAM, ROM, EEPROM, flash memory orother memory technology, CD-ROM, digital versatile disks (DVD) or otheroptical disk storage, magnetic cassettes, magnetic tape, magnetic diskstorage or other magnetic storage devices, or any other medium that maybe used to store the desired information/data and which may be accessedby the general purpose computing device. Communication media includescomputer-readable instructions, data structures, program modules orother data in a modulated data signal such as a carrier wave or othertransport mechanism and includes any information delivery media. Theterm “modulated data signal” means a signal that has one or more of itscharacteristics set or changed in such a manner as to encode informationin the signal. For example, communication media may include wired mediasuch as a wired network or direct-wired connection and wireless mediasuch as acoustic, RF, infrared, and/or other wireless media, or somecombination thereof. Computer-readable media may be embodied as acomputer program product, such as software stored on computer storagemedia.

The main memory 204 includes computer storage media in the form ofvolatile/nonvolatile memory such as read only memory (ROM) and randomaccess memory (RAM). A basic input/output system (BIOS), containing thebasic routines that help to transfer information between elements withinthe general purpose computing device (e.g., during start-up) istypically stored in ROM. RAM typically contains data and/or programmodules that are immediately accessible to and/or presently beingoperated on by processor 202. Further, data storage 206 stores anoperating system, application programs, and other program modules andprogram data.

The data storage 206 may also include other removable/non-removable,volatile/nonvolatile computer storage media. For example, data storage206 may be: a hard disk drive that reads from or writes tonon-removable, nonvolatile magnetic media; a magnetic disk drive thatreads from or writes to a removable, nonvolatile magnetic disk; and/oran optical disk drive that reads from or writes to a removable,nonvolatile optical disk such as a CD-ROM or other optical media. Otherremovable/non-removable, volatile/nonvolatile computer storage media mayinclude magnetic tape cassettes, flash memory cards, digital versatiledisks, digital video tape, solid state RAM, solid state ROM, and thelike. The drives and their associated computer storage media providestorage of computer-readable instructions, data structures, programmodules and other data for the general purpose computing device 200.

A user may enter commands and information through a user interface 240(displayed via a monitor 260) by engaging input devices 245 such as atablet, electronic digitizer, a microphone, keyboard, and/or pointingdevice, commonly referred to as mouse, trackball or touch pad. Otherinput devices 245 may include a joystick, game pad, satellite dish,scanner, or the like. Additionally, voice inputs, gesture inputs (e.g.,via hands or fingers), or other natural user input methods may also beused with the appropriate input devices, such as a microphone, camera,tablet, touch pad, glove, or other sensor. These and other input devices245 are in operative connection with the processor 202 and may becoupled to the system bus 201, but may be connected by other interfaceand bus structures, such as a parallel port, game port or a universalserial bus (USB). A monitor 260 or other type of display device is alsoconnected to the system bus 201. The monitor 260 may also be integratedwith a touch-screen panel or the like.

The computing device 200 may be implemented in a networked orcloud-computing environment using logical connections of a networkinterface 203 to one or more remote devices, such as a remote computer.The remote computer may be a personal computer, a server, a router, anetwork PC, a peer device or other common network node, and typicallyincludes many or all of the elements described above relative to thegeneral purpose computing device. The logical connection may include oneor more local area networks (LAN) and one or more wide area networks(WAN), but may also include other networks. Such networking environmentsare commonplace in offices, enterprise-wide computer networks, intranetsand the Internet.

When used in a networked or cloud-computing environment, the computingdevice 200 may be connected to a public and/or private network throughthe network interface 203. In such embodiments, a modem or other meansfor establishing communications over the network is connected to thesystem bus 201 via the network interface 203 or other appropriatemechanism. A wireless networking component including an interface andantenna may be coupled through a suitable device such as an access pointor peer computer to a network. In a networked environment, programmodules depicted relative to the general purpose computing device, orportions thereof, may be stored in the remote memory storage device.

Computing System

Referring to FIG. 4, in some embodiments a computer-implementedframework for enterprise pricing as described herein may be implementedat least in part by way of a computing system 300. In general, thecomputing system 300 may include a plurality of components, and mayinclude at least one computing device 302, which may be equipped with atleast one or more of the features of the computing device 200 describedherein. As indicated, the computing device 302 may be configured toimplement an optimized pricing model 304 which may include one or moreof a solution algorithm 306 for generating optimized pricing asdescribed herein. Aspects of the optimized pricing model 304 may beimplemented as code and/or machine-executable instructions executable bythe computing device 302 that may represent one or more of a procedure,a function, a subprogram, a program, a routine, a subroutine, a module,a software package, a class, or any combination of instructions, datastructures, or program statements related to the above model/s andmethods. A code segment of the optimized pricing model 304 may becoupled to another code segment or a hardware circuit by passing and/orreceiving information, data, arguments, parameters, or memory contents.Information, arguments, parameters, data, etc. may be passed, forwarded,or transmitted via any suitable means including memory sharing, messagepassing, token passing, network transmission, or the like.

In other words, aspects of the optimized pricing model 304 may beimplemented by hardware, software, firmware, middleware, microcode,hardware description languages, or any combination thereof. Whenimplemented in software, firmware, middleware or microcode, the programcode or code segments to perform the necessary tasks (e.g., acomputer-program product) may be stored in a computer-readable ormachine-readable medium, and a processor(s) associated with thecomputing device 302 may perform the tasks defined by the code; suchthat the computing device 302 is configured via the aforementionedhardware and software components to perform the optimized pricefunctionality described herein.

As further shown, the system 300 may include at least one internetconnected device 310 in operable communication with the computing device302. In some embodiments, the internet connected device 310 may providepricing and market data 312 to the computing device 302 for trainingpurposes or real world pricing optimization. The internet connecteddevice 310 may include any electronic device capable ofaccessing/tracking pricing and market data such over a predeterminedperiod of time. In addition, the system 300 may include a clientapplication 320 which may be configured to provide aspects of theoptimized pricing model 304 to any number of client devices 322 via anetwork 324, such as the Internet, a local area network, a wide areanetwork, a cloud environment, and the like.

Example embodiments described herein may be implemented at least in partin electronic circuitry; in computer hardware executing firmware and/orsoftware instructions; and/or in combinations thereof. Exampleembodiments also may be implemented using a computer program product(e.g., a computer program tangibly or non-transitorily embodied in amachine-readable medium and including instructions for execution by, orto control the operation of, a data processing apparatus, such as, forexample, one or more programmable processors or computers). A computerprogram may be written in any form of programming language, includingcompiled or interpreted languages, and may be deployed in any form,including as a stand-alone program or as a subroutine or other unitsuitable for use in a computing environment. Also, a computer programcan be deployed to be executed on one computer, or to be executed onmultiple computers at one site or distributed across multiple sites andinterconnected by a communication network.

Certain embodiments may be described herein as including one or moremodules. Such modules are hardware-implemented, and thus include atleast one tangible unit capable of performing certain operations and maybe configured or arranged in a certain manner. For example, ahardware-implemented module may comprise dedicated circuitry that ispermanently configured (e.g., as a special-purpose processor, such as afield-programmable gate array (FPGA) or an application-specificintegrated circuit (ASIC)) to perform certain operations. Ahardware-implemented module may also comprise programmable circuitry(e.g., as encompassed within a general-purpose processor or otherprogrammable processor) that is temporarily configured by software orfirmware to perform certain operations. In some example embodiments, oneor more computer systems (e.g., a standalone system, a client and/orserver computer system, or a peer-to-peer computer system) or one ormore processors may be configured by software (e.g., an application orapplication portion) as a hardware-implemented module that operates toperform certain operations as described herein.

Accordingly, the term “hardware-implemented module” encompasses atangible entity, be that an entity that is physically constructed,permanently configured (e.g., hardwired), or temporarily configured(e.g., programmed) to operate in a certain manner and/or to performcertain operations described herein. Considering embodiments in whichhardware-implemented modules are temporarily configured (e.g.,programmed), each of the hardware-implemented modules need not beconfigured or instantiated at any one instance in time. For example,where the hardware-implemented modules comprise a general-purposeprocessor configured using software, the general-purpose processor maybe configured as respective different hardware-implemented modules atdifferent times. Software may accordingly configure a processor, forexample, to constitute a particular hardware-implemented module at oneinstance of time and to constitute a different hardware-implementedmodule at a different instance of time.

Hardware-implemented modules may provide information to, and/or receiveinformation from, other hardware-implemented modules. Accordingly, thedescribed hardware-implemented modules may be regarded as beingcommunicatively coupled. Where multiple of such hardware-implementedmodules exist contemporaneously, communications may be achieved throughsignal transmission (e.g., over appropriate circuits and buses) thatconnect the hardware-implemented modules. In embodiments in whichmultiple hardware-implemented modules are configured or instantiated atdifferent times, communications between such hardware-implementedmodules may be achieved, for example, through the storage and retrievalof information in memory structures to which the multiplehardware-implemented modules have access. For example, onehardware-implemented module may perform an operation, and may store theoutput of that operation in a memory device to which it iscommunicatively coupled. A further hardware-implemented module may then,at a later time, access the memory device to retrieve and process thestored output. Hardware-implemented modules may also initiatecommunications with input or output devices.

It should be understood from the foregoing that, while particularembodiments have been illustrated and described, various modificationscan be made thereto without departing from the spirit and scope of theinvention as will be apparent to those skilled in the art. Such changesand modifications are within the scope and teachings of this inventionas defined in the claims appended hereto.

What is claimed is:
 1. A method of computer implemented enterprisepricing, comprising: utilizing a processor in communication with atangible storage medium storing instructions that are executed by theprocessor to perform operations comprising: defining a priceoptimization function taking as input at least a vector of prices of aset of new products; redefining a term for a profit in the priceoptimization function as a function of choice probabilities of the setof new products; generating a value of concavity of the profit; andemploying a bisection search algorithm to solve for an optimal price andan optimal profit of the set of new products by: generating an upperlimit of the price optimization function, generating a lower limit ofthe price optimization function; averaging across the upper limit andthe lower limit to find an average of the upper limit and the lowerlimit, and inputting the average of the upper limit and the lower limitinto the price optimization function.
 2. The method of claim 1, whereinan attribute value of the set of new products is predefined.
 3. Themethod of claim 1, wherein a value of marginal utility of an attributevalue of the set of new products depends upon a price of the set of newproducts.
 4. The method of claim 1, wherein a value of marginal utilityof the price of the set of new products depends upon an attribute valueof the set of new products.
 5. A method of computer implementedenterprise pricing, comprising: utilizing a processor in communicationwith a tangible storage medium storing instructions that are executed bythe processor to perform operations comprising: defining a first valueof customer utility reflecting a composite utility value of a product;defining a second value of customer utility reflecting an attribute;defining a product cost function inputting at least the first value ofutility; defining an attribute optimization problem inputting at least avector of attribute values for a new product, the product cost function,and the second value of customer utility and outputting a maximum valueof an utility function; and employing a bisection search algorithm tosolve the attribute optimization problem by: generating a first upperlimit of the attribute optimization problem, generating a first lowerlimit of the attribute optimization problem, averaging across the firstupper limit and the first lower limit to find a first average of thefirst upper limit and the first lower limit, inputting the average ofthe first upper limit and the first lower limit into the utilityfunction, generating a second upper limit of the utility function,generating a second lower limit of the utility function, averagingacross the second upper limit and the second lower limit to find asecond average of the second upper limit and the second lower limit, andinputting the first average and the second average into the attributeoptimization problem.
 6. The method of claim 5, wherein a price value ofthe product is exogenous.
 7. The method of claim 5, wherein a value ofmarginal utility of an attribute value of the product depends upon theprice value of the product.
 8. The method of claim 5, wherein a value ofmarginal utility of the price value of the product depends upon theattribute value of the product.
 9. The method of claim 5, wherein anattribute value of the product increases with a value of pricesensitivity of the product.
 10. A method of modeling interaction betweena product attribute and a product price, comprising; utilizing aprocessor in communication with a tangible storage medium storinginstructions that are executed by the processor to perform operationscomprising: employing a multinomial logit model to generate a value ofpurchase probability by: defining a utility function inputting anattribute value, a price of a product, an observable independent utilityterm, and a randomly generated noise term; redefining the utilityfunction into a first form considering a marginal disutility of theprice of the product; redefining the utility function into a second formconsidering a marginal utility of the attribute value; defining a set ofexisting products and a set of new products; setting a no-purchaseoption having a utility value of zero; and generating a purchaseprobability value of the set of new products, a purchase probability ofthe set of existing products, and a no-purchase probability each asfunctions of the utility function, the first form of the utilityfunction, and the second form of the utility function.
 11. The method ofclaim 10, wherein the randomly generated noise term is a Gumbel randomvariable.
 12. The method of claim 10, wherein the marginal disutilitydecreases as the attribute value decreases.
 13. The method of claim 10,wherein the marginal utility increases as the price increases.
 14. Themethod of claim 10, further comprising employing the multinomial logitmodel to model demand of the product.
 15. The method of claim 10,further comprising employing the multinomial logit model to optimize theprice of the product and the attribute value of the product.
 16. Themethod of claim 10, wherein the marginal utility of the attribute valuedepends upon the price.
 17. The method of claim 10, wherein the marginalutility of the price depends upon the attribute value.
 18. The method ofclaim 10, wherein the attribute value is exogenous.
 19. The method ofclaim 10, wherein the attribute value decreases with a value of pricesensitivity.
 20. The method of claim 10, wherein the attribute value ismulti-dimensional.